To make brown butter, just take your butter and cut it in small cubes. Throw it into a saucepan on the stove, and melt it over medium heat.
Once the butter has melted, keep it on the stove and continue heating it. It will get bubbly and splatter around the saucepan a bit. (That’s why it’s good to do this in a saucepan with tall sides.)
I like to jostle my saucepan on the stovetop every so often once it gets to the splattery stage, just to keep the liquid moving around a bit. While you could do the same thing with a spoon or whisk, you risk losing some of the butter every time you take that utensil in and out. So I try to move the pan enough to swirl the liquid butter every few minutes.
After 5 minutes or so, the melted butter will have lost a lot of the creamy yellow color that it started with, and begin to look almost clear. It’s at this point that you know the butter is almost browned. It’s also at this point that I begin to gently, constantly shake the saucepan–just enough to keep that butter in motion.
When tiny brown flecks start to appear in the center of the pan, you’ve got brown butter.
Remove the pan from the stovetop right away, so that the butter won’t burn. It’s also helpful to pour the butter out into a heatproof bowl at this point. Make sure you use a spatula to scrape all the little brown bits out of the saucepan–those bits are packed with flavor!
Allow the butter to cool for about 30-35 minutes. (This is important!)