Method 1: Theoretical
According to this site, “commercial butter is 80â€“82 percent milk fat, 16â€“17 percent water, and 1â€“2 percent milk solids other than fat.” We have a little salt in there too.
Presuming standard atmosphere conditions, the water will boil at 100C (212F). The fats and solids should raise that temperature a bit as boiling point elevation can be accomplished by adding compounds to a solvent. Wikipedia reports that milk solids brown around 150C. My first idea for a crude estimate would be to calculate the boiling point as a percentage of composition. ~80% milk fats and ~20% water = 132C. This however is probably not realistic. It’s more likely that the water will begin to boil first, then the milk fats will brown.
Temperature estimation for that butter in the pan: 115C – 120C
Method 2: Anecdotal
On Tuesday, I splashed some boiling butter on my face. I now have two blistering welts on my forehead.
Temperature estimation for that butter in the pan: freakin’ hot.