

Unfortunately, scrambled eggs are one of those things that are a bit harder to cook in stainless. If you’ve been using oil, switching to butter can help with some of the sticking, but you might end up using more butter than you’d like and some heat management is still necessary. Honestly, a lot of people keep a single nonstick frying pan just for things like eggs, seared fish, or sauteed tofu, so don’t feel pressured to make it work on stainless.
If you really feel compelled to move off Teflon completely, a carbon steel or cast iron pan will be much better suited to replacing nonstick for the instances where you’re getting excessive sticking, at the cost of needing to avoid acids in the pan. But if I were you, I’d just keep using my Teflon pans until there’s any flaking or chipping before making up my mind.
Good luck finding an article on point for this sort of thing. I looked for a bit to try to find something to link here, but couldn’t find anything that directly addressed cooking oil polymers. I just wanted to say that you won’t necessarily be eating burnt fat. Seasoning cast iron/carbon steel isn’t necessarily about burning fat onto the surface of the pan, since what you’re trying to do is create a polymerized layer of cooking oil on the surface of the pan. Polymerization can occur well below the smoke point of an oil, you just apply heat to speed up the process. Applying too much heat can actually be counterproductive, since the polymer will carbonize, but you’ll probably never reach this point unintentionally without leaving a pan on the burner unattended. Under normal cooking conditions, any heat degradation of the polymer layer will be made up for with the fresh cooking oil that you’ve used, refreshing the seasoning. If you’re going to use cast iron or carbon steel, you should thoroughly clean any burnt on bits after cooking with dish soap. Modern dish soaps don’t have lye, so you won’t be doing any damage to the seasoning and you’ll only be left with polymerized oil, not burnt fat.
If you do decide to do more research and find something on point, please do share! I wasn’t able to find anything that explicitly pointed towards it being unhealthy, and I’m alright with making a somewhat informed assumption of the risk.