Just a random thought - while many recipes can be made in pots & pans of different shapes and different materials without really affecting the outcome, browning butter easily without burning it might be an exception. I don’t know what you’ve browned butter in (i.e. you may have solved this part already), but if the stuff has frustrated you it’s possible that some little thing like that is causing more than its share of trouble.
Conventional wisdom for most foods is that something that includes a thick layer of aluminum or a thick layer of copper is the most “forgiving” in preventing burnt food compared to plain stainless steel or basically anything. AFAIK this is because they heat more evenly with fewer hot spots. Bare aluminum should also be easier than bare stainless if they’re both thin. Cast iron usually stores too much heat for this (keeps on cooking after the heat is off), but somehow I don’t imagine people ordinarily using cast iron saucepans anyway.
And then again, maybe browning butter goes against conventional wisdom - I wouldn’t know. Even just making a change for the sake of change might create a browned-butter revolution.
This post brought to you by “Aluminum Doesn’t Suck After All”