Cut butter into tablespooon-sized pats and melt over medium heat in a small heavy-bottomed skillet. Choose a skillet that's light in color (stainless or white/ivory enameled) so you can monitor the color of the butter as it browns. When foaming subsides, continue cooking, swirling the pan frequently, until the milk solids are golden brown and the butter has a nutty aroma. (Watch it closely! It can burn quickly.)
Immediately pour brown butter into a bowl, leaving behind as much of the dark solids in the pan as possible. (You can also pour the brown butter through a fine mesh strainer, if desired.) Set aside to cool just slightly while you sift and measure your powdered sugar.
Gradually add the powdered sugar to the brown butter, whisking to combine. Whisk in vanilla extract and enough milk or half and half to thin the glaze to your desired consistency for dipping, drizzling, or pouring.
SeeFAQ's above for make ahead and glazed cake storage instructions.
Spiked Brown Butter Glaze: Substitute 1/2 to 1 tablespoons of your favorite liquor or liqueur (such as bourbon, rum, or Grand Marnier) for some of the milk or half and half. Citrus Brown Butter Glaze: Omit vanilla extract. Substitute 1-2 tablespoons lemon juice or orange juice for the milk or half and half. Stir in 1 teaspoon finely-grated citrus zest.Salted Brown Butter Glaze: Use salted butter, or add additional kosher or sea salt, to taste.