While your potatoes are cooking, place 1 1/2 sticks of butter (12 tablespoons total for a generous amount of brown butter; see notes) in a small or medium heavy pot, such as a stainless steel one. Try not to use a dark pot, or it will be hard to notice the color of the butter as it cooks. Place the pot over medium heat on the stove and let the butter melt. As the butter starts to bubble, lower the heat and stir the butter occasionally as it foams (watch the pot constantly so the butter doesn't burn.) Cook the butter just until it turns a caramel color (you can use your spoon or spatula to push away some bubbles to peak at the color of the butter.) Immediately turn off the heat and place the pot on a cool burner. Transfer the butter to a bowl so it doesn't continue cooking.
For the milk, start with 1/2 cup, stir it in and see if you like the consistency. You can add up to another 1/2 cup of milk if needed.
For the browned butter, you can get away with just one stick of butter (8 tablespoons) and have just enough...but I put 12 tablespoons in the recipe so there is plenty to go around.