Put butter in a light-colored pan over medium heat (it should be light color so it's easy to see when it's browning). Constantly stir butter as it melts. Once melted, it should start to bubble and in just a minute or two, you'll start seeing brown flecks on the bottom of the saucepan and the butter will start smelling nutty.
Remove from heat and continue to stir for another 30 seconds. Pour into a small heat-proof bowl. If you used unsalted butter, stir in a pinch salt here.
Immediately stir in brown sugar until well combined.