Preheat oven to 375 degrees. Line cookie sheets with parchment paper. Set aside.
In the bowl of a stand mixer, cream the butter and sugar together until light and fluffy. Add in the eggs, vanilla, and flavor extract and beat until combined.
Slowly add the flour, stirring just until combined. Turn dough out onto a lightly floured work surface and use your hands to form into a disk. Roll the dough to 1/8 to 1/4-inch thick and cut with lightly floured cookie cutters. Place cookies on the prepared baking sheets.
Bake cookies for 10-12 minutes, or just until the edges barely start to turn golden. Remove from the baking sheets to a rack to cool before icing.
For the icing, beat the butter with an electric mixer until creamy. Beat in the powder sugar, then add the milk and flavor extract. Beat until combined, adding more milk or more powdered sugar as needed to achieve desired spreading consistency. If coloring, beat in food coloring. Spread onto cookies and allow to set before storing.
Notes
*Please use a high-quality flavor extract. My favorite for these cookies is Nielsen-Massey Pure Almond Extract or Nielsen-Massey Pure Lemon Extract.