In a bowl, with an electric mixer on high speed, beat butter and sugar until smooth and light. Beat in eggs, lemon peel, lemon juice, milk, and vanilla until well blended, scraping down sides of bowl as necessary.
In another bowl, stir together flour, baking powder, and salt. Stir or beat into butter mixture until blended. Wrap dough in plastic wrap and chill until firm, at least 2 hours, or up to 3 days.
Unwrap dough and place on a lightly floured surface.
Roll dough out, a portion at a time, to about 3/8 inch thick.
Cut with a 3- or 4-inch round cutter and place cookies slightly apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake cookies in a 350 oven until edges are golden, 10 to 14 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. With a wide spatula, transfer cookies to wire racks to cool completely.
Powdered sugar icing. Controlling consistency of icing is simple. If it's too thick, add a few drops of water; if it's too thin, gradually add more powdered sugar. To tint, add food coloring a few drops at a time.
For spreading: In a bowl, stir 3 cups powdered sugar, 1 teaspoon vanilla, and 1/4 cup water until smooth. Tint with food coloring as desired.
Spread icing onto cool cookies with a metal spatula.
Let stand until icing is firm, about 1 1/2 hours. Makes about 1 cup.
For piping: In a bowl, stir 2 cups powdered sugar, 1/2 teaspoon vanilla, and 1 1/2 tablespoons water until smooth. Tint with food coloring as desired. Spoon into a pastry bag fitted with a 1/8-inch tip or a heavy 1-quart zip-lock plastic bag. Seal plastic bag and squeeze icing into a corner; cut a tiny opening at corner of bag to squeeze icing out.
Let cookies stand until icing is firm, about 5 minutes. Makes about 3/4 cup.
