Browned butter is made by cooking butter until the milk solids in the butter begin to toast and turn a lovely brown color. This is typically done on the stove in a sauce pan, over medium or medium-low heat where it can be watched closely for color and doneness. The solids in the butter can go from toasted to burnt very quickly. This is a fine way to brown butter, just as using a double boiler is a fine way to melt chocolate. Now, chocolate can be melted more quickly in the microwave as long as you keep an eye on it to prevent it from burning. The same is true for butter: it can easily be browned in the microwave.
Put your butter into a large, microwave safe dish and heat it in 1-minute intervals in the microwave, checking after the second minute to look for brownness bubbling away at the bottom of the dish. Try using a 4-cup size bowl for 1 cup or less of butter because the butter will puff up and expand as it cooks, and will definitely bubble over the top of the bowl if you aren’t using one that is large enough. The whole process will take about 3-4 minutes, depending on how much butter you’re cooking. You’ll see lots of brown flecks floating in the butter when it is done (these are the milk solids). Once the butter is cooled, you’re ready to incorporate it into any recipe.