How do I determine roasting temperature as a function of time? [duplicate]

Question

TL;DR

If it takes 90 minutes to roast a leg of lamb at some temperature (perhaps somewhere from 350 to 450 °F?), and it takes 840 minutes to slow-roast a leg of lamb of the same exact weight at some other temperature (possibly 175 to 275 °F?), what is the temperature setting that one should use if one wants the leg of lamb to spend 420 minutes in the oven? Is it suitable to calculate the temperature using linear interpolation?

The mid-length story:

The purpose of the question is to avoid the tyranny of recipes, where I have to go to my tablet, computer, or book shelves to do anything. I'd like to understand the ideas (in this case, for roasting lamb or beef) well enough to be able to cook without the handholding of recipe authors.

In this era of internet-sourced recipes, recipes seem to be, by and large, copied, or inspired (possibly even sometimes plain plagiarized). For example, search online for "roast leg of lamb" and you will find a hundred pundits claiming to provide a recipe. Search for "slow roast leg of lamb" and you will find more results. But these recipes are not gospel. We should be able to do anything we want, so long as it's flavorful and healthy. Is there such a thing as semi-slow, average-slow, and rather-fast-than-slow roasting, or are we stuck with just "roasting" and "slow roasting"?

The long story:

Cooking a whole lamb, starting by defrosting it, will have to remain a 5- or 10-year project, when I will know that I can nail every detail right, and when I will know that a 40-or-so-large family reunion will happen, if they're ever in the same place at the same time.

At this time I have the much more modest objective of properly cooking a leg of lamb, defrosted for 36 hours in the fridge

and smeared with mustard, rosemary, and garlic

My intention was to leave it in the oven overnight at 275 °F / 135 °C. Slow roasting is much more of a foolproof method, because I don't have to calculate precise minutes-per-weight numbers. If it's in the oven two hours more or an hour less, it's still perfectly alright. (Yes, yes, no need to lecture me about food safety; I do make sure that the internal thermometer reading reaches whatever the thermometer says.)

Here is the problem: I really don't want the flavor, the texture, or, as mentioned above, the time sensitivity of roasting at 375 °F / 190 °C. I would like the fall-off-the-bone tenderness of 275 °F / 135 °C, but.

I was much too tired (tennis for just one hour did it) to think about standing on my feet for the 5-10 minutes that it takes to prepare the roast before going to bed. I'm starting to roast early in the morning instead, for guests who will arrive 420 minutes after the roast went to the oven.

If I need 90 minutes at 375 °F / 190 °C and 840 minutes (14 hours) at 275 °F / 135 °C, I am speculating that there is a precise temperature that I can use for 420 minutes, and the roast will be just perfect when the guests arrive. (Then the roast will rest outside during chatting and appetizers.)

How do I determine roasting temperature as a function of time? Would, for example, linear interpolation

375 + (420-90) * (275-375) / (840-90) = 331 °F

make sense?

N.B.: This might be a difficult question, in the sense that cookbooks do not discuss it. Feel free to "throw me a bone", figuratively speaking, and make a suggestion that I can try. If the roast is under-cooked, I can always use one of various remedies and apologize that I didn't get it quite right. The guests will be gracious. But if a better answer comes along later, it's fairer, and better fitting for a reference site, that I update the "correct" answer. Cooking is an art form anyway. There isn't just "right" and "wrong".

Answer 1

I have not seen this discussed in terms of oven cookery, however, it is certainly a part of the conversation when cooking with sous vide. Douglas Baldwin, noted expert, has a guide. Here, I have linked his appendix, "The mathematics of sous vide." Now, I am certain that you CAN NOT use this guide for oven cookery, because in the oven, you have a significant amount of evaporative cooling happening, which I think would dramatically change these calculations (I'm no mathematician). However, I do include it to point out that the size and shape of your food must be taken into account. You can't just consider time and temperature.

How do you know you need X minutes at Y degrees? For how much lamb and bone?

I googled "slow roasted leg of lamb." There are plenty of recipes, I don't see anyone cooking them for 14 hours. I think you can achieve the texture you want in much less time. So, I doubt you need to be as precise as you are suggesting. You might err on the side of your roast being finished early, rather than late, but even if it were finished well before your guests arrived, that is workable for a delicious meal.