Step 1In the bowl of a stand mixer fitted with the paddle attachment, or in a large mixing bowl using a hand mixer, cream the butter and sugar until well blended, 3 to 5 minutes. Beat in the eggs, one at a time, until combined, then beat in the vanilla extract.
Step 2In a separate bowl, whisk together the flour, baking powder and salt. Beat the dry ingredients into the butter mixture to form a dough, then cover and refrigerate the dough until well-chilled, preferably overnight.
Step 3Heat the oven to 350 degrees. On a lightly floured surface, roll the dough out to a thickness of one-fourth inch. Use various cookie cutters to cut out desired shapes, and space the cookies a couple of inches apart on ungreased cookie sheets.
Step 4Bake the cookies until the edges are lightly browned and the cookies are set, 10 to 12 minutes (time will vary depending on the size and shapes of the cookies), rotating the cookies halfway through for even coloring. Cool completely before frosting.
Step 1In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl using a hand mixer, cream together the butter and sugar. Beat in milk as needed to thin; the icing should be creamy but not runny. Add lemon extract to taste. This makes approximately 21/4 cups buttercream. Color the buttercream with food coloring as desired to decorate the cookies, then set the cookies aside to dry before serving.