Blend 2 cups butter and sugar together; stir in eggs and 2 teaspoons vanilla. Add flour; mix until well blended. Shape into a ball; cover and chill for 4 hours to overnight. Roll out dough 1/4-inch thick on a lightly floured surface; cut out with cookie cutters as desired. Arrange cookies on lightly greased baking sheets. Bake at 350 degrees for 8 to 10 minutes, until golden; cool.
To make Frosting: Combine powdered sugar, milk, remaining butter (melted), remaining vanilla, lemon juice and food coloring, if using, in a medium bowl. Beat with an electric mixer on low speed until smooth. Frost cookies when cool.