In a medium bowl, 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. In another bowl, stir together flour, baking powder and salt. Stir into butter mixture until blended. Wrap dough in plastic wrap and chill until firm, at least 2 hours. Preheat oven to 350 degrees F. Grease cookie sheet or line with parchment. Roll dough out on a floured surface, a portion at a time, to about 3/8 -inch thick. Cut out cookies and place about an inch apart on cookie sheet. Bake cookies until edges are golden, 10 to 14 minutes. Transfer cookies to wire racks to cool completely. To make icing for spreading: In a bowl, stir 3 cups confectioners’ sugar, 1 tsp vanilla, and ¼ 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 1½ hours.
Makes about 1 cup. To make icing for piping: In a bowl, stir 2 cups confectioners’ sugar, ½ tsp vanilla, and 1½ Tbsp water until smooth. Tint with food coloring as desired. Spoon into a pastry bag fitted with a ¹/8 inch tip or a zip-lock bag. Decorate as desired. Let cookies stand until icing is firm, about 5 minutes. Makes about ¾ cup.