Melt the butter in a small saucepan over medium heat. Stir constantly while cooking just until the butter starts to turn golden brown. This will take about 4-6 minutes. Watch it closely because it can burn easily once the solids separate. Cool completely.
Combine the browned butter, powdered sugar, cinnamon and vanilla in a small mixing bowl. Beat on low speed until combined. Add milk 1 tablespoon at a time until desired consistency. Frost the cooled cake as desired. Sprinkle with chopped pecans if using.