In a medium saucepan, melt butter over medium heat until a deep golden brown. Place browned butter in fridge or freezer to chill until completely solidified.
With an electric mixer, beat browned butter with sugar until completely creamed. Beat in zests, vanilla extract, milk, and flour until a soft dough forms. Wrap dough in plastic wrap and chill 1 hour until completely cold.
On a lightly floured surface, roll dough out to 1/2-in thickness. Use a 3x1-in cookie cutter to cut out cookies. Place cookies 2-in apart on parchment paper-lined baking sheets.
Bake cookies at 350F 12-15 minutes until edges begin to turn golden (if baking in batches, refrigerate un-baked cookies while first batch bakes--this dough needs to be kept cold!)
Cool cookies 10 minutes on baking sheet, then transfer to a wire rack to cool completely.