Cooking pareve food in a dairy pot for consumption in a meat meal (or vice versa) Rabbanit Chanital Ofan

This question precisely is actually discussed in the Gemara. Hullin 111b addresses the case of fish cooked in a meat pot, and asks whether it can be eaten with kotah – a dairy mixture of milk and breadcrumbs or barley flour.[1] Rav prohibited the fish for consumption, but Shmuel allowed it. According to Rav, the fish absorbed flavor from the meat pot, whereas according to Shmuel the flavor transferred is too weak to be of any concern. The term used in the context of this discussion is נותן טעם בר נותן טעם (noten ta’am bar noten ta’am, commonly referred to with the abbreviated nat bar nat), indicating ‘secondary taste’ – food that absorbed the taste indirectly from the pot, and not directly from the meat. The Gemara accepts Shmuel’s position, and concludes: “fish cooked in a meat pot may be eaten with kotah.” It is important to note that the nat bar nat rule only applies to the prohibition of mixing meat and dairy, since each of the components independently is kosher, and not inherently prohibited. The weak flavor absorbed into the fish from the meat pot also does not create any prohibition, since the fish is pareve (Rashi s.v. נ”ט בר נ”ט). Arukh ha-Shulhan,[2] citing the Ran,[3] states that the secondary flavor cannot create a prohibition when it is not inherently prohibited. Conversely, this rule would never apply in the case of a non-kosher pot, since the initial flavor (even if weak) absorbed would render the fish prohibited for consumption.

Rishonim debated the applicability of the Gemara’s allowance on several levels:

To what is the Gemara referring when it discusses cooking in a pot (using the unusual phrase עלו בקערה)? Does this apply equally to cooking, roasting and baking, or only to boiling in water?
Is this allowance a prioiri (lechatchila) – that is, can fish be cooked in a meat pot with the intention of serving it with dairy, or only post factum (be-dieved) – whereby in the event that fish was already cooked in a meat pot (without intention of serving it with dairy) – it can still be served with dairy?
Does the leniency also apply to a pot that was used in the last 24 hours (not ben yomo)?

Three approaches to this sugya can be found in the Rishonim:

According to the Rambam,[4] Rashba,[5] Ran,[6] Rabbenu Tam,[7] RY,[8] Raavia,[9] Rashi, and others, the Gemara’s allowance is lechatchila, and no distinction should be made between cooking and roasting: in all cases of a pareve dish cooked in a meat pot, the food may be consumed with dairy. The Ran and Raavia explain that had there been a distinction between roasting and cooking, Rav would have noted that this was the case; the Gemara used cooking as an example, and there is no reason to assume roasting is in a different category. According to Ran and Raavia, Shmuel’s allowance applies to both cooking and roasting. According to the Ran, the case description emphasizes that flavor is transferred twice (from the meat to the dish, and from the dish to the fish) – in which case certainly the allowance should apply to roasting as well.
Rivan[10] understands based on Rashi that the term עלו is deliberate, indicating that food placed in a meat dish may be eaten with dairy – but if the food was cooked or roasted this would not be the case. It is important to emphasize that this is a restriction for a dish that was used in the last 24 hours (ben yomo). If the pot was not used in the last 24 hours, it is considered noten ta’am lifgam (imparts a foul taste), and in this case the food may be consumed with dairy bedieved. Tosfot[11] cites the Rivan’s ruling and reinforces his position based on the prohibition to eat radishes cut with a meat knife together with dairy.[12] Rivan argues that this is an application of nat bar nat in the case of cutting a spicy vegetable, which cannot possibly be a more stringent case than cooking.[13]
Sefer ha-Teruma,[14] Rosh,[15] Semak,[16] and Semag,[17] offer an offshoot of the Rivan’s approach. They distinguish between roasting, in which direct flavor is transferred from the pot, and cooking, where the transfer of flavor includes three stages: from the dish to the water; from the water to the pareve food; and from the pareve food to the dairy food. Therefore, the allowance only applies to the case of cooking, but fish roasted in a dry meat pot cannot be eaten with dairy.[18] Rivan may adopt this approach and only prohibit when the fish was roasted in a pot, but when cooked in water it should be permitted to be eaten with milk/dairy according to all approaches.[19]

The provision of kli she’eino ben yomo (a dish that was not used in the last 24 hours)

We can expand on the Rivan’s approach, which the Rema adopts in practice, at least lechatchila. The Mordechai cites the Rivan (in the name of Tosfot Shanz) and adds that when the food was cooked in water it should be obvious that it can be eaten with dairy, since there are three levels of separation in the transfer of flavor (the flavor from the pot, the flavor from the water, and the flavor from the fish). According to the Mordechai only two main approaches exist: the Rambam and his school of thought who permit nat-bar-nat in all instances, or the Rivan and his school of thought who limit this allowance.

On a practical plane there are two approaches among poskim regarding the status of food cooked in water (referred to as טעם שלישי). There is a general agreement that this food may be eaten with dairy, but poskim disagree whether this allowance is lechatchila or bedieved. Semak[20] argues that the Rivan’s allowance only applies after the fact; conversely, Sefer ha-Teruma and others believe food may be cooked in water in a meat pot lechatchila for consumption with dairy. Yam Shel Shlomo[21]  cites the Sefer ha-Teruma but testifies that it is customary to be stringent with regard to a ben yomo dish used in the last 24 hours, in accordance with Semak. According to Ohalei Yehuda[22] there are two parts to this debate: Sefer ha-Teruma believes food cooked in water is permissible lechatchila even in a ben yomo dish, while the Semak holds that this is only permitted bedieved even in a dish that was not used in the last 24 hours. Maharshal[23] ruled according to Semak that since the custom is to be stringent there should be no distinction between ben yomo and a dish that was not used in the last 24 hours, even when cooking with water.

Even according to the more stringent approach, in which the food may not be consumed with dairy – the food itself is not meat in status – and certainly one may subsequently eat dairy.