During her "Food & Wine" demo, Stewart described how to clarify butter in some very simple terms: "You pour off the melted butter, the butter fat, and that's your clarified butter."
Since I wasn't familiar with the process before this experiment, I got a little more clarification (see what I did there) with the help of this step-by-step article from Culinary Hill.
"Melt the butter over low heat," author Meggan Hill writes. "If the butter boils, the milk solids get dispersed throughout the fat and you won't be able to skim them off."
"Skim off the foamy milk solids that rose to the top," she continues. "What you're left with is pure butterfat."
Sounds easy enough! So I added knobs of butter to a small saucepan, turned the stove to low heat, and waited for the magic to happen.