With a zester, remove lime peel from lime; set aside. In a large mixing bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, baking powder, and salt. Beat until combined, scraping sides of bowl occasionally. Beat in egg, the 1 tablespoon lime juice, and vanilla until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour and 1 teaspoon of the lime peel. Divide dough in half. Cover and chill at least 2 hours or up to 3 days.

Preheat oven to 350 degrees F. On a well-floured surface, roll half the dough at a time to 1/8-inch thickness. Using a 2- or 3-inch flower-shape cookie cutter, cut dough in shapes. Place cookies 1 inch apart on an ungreased cookie sheet.

Bake for 8 to 10 minutes for the 2-inch cookies or 10 to 12 minutes for the 3-inch cookies or until edges just start to brown. Transfer to a wire rack to cool.

For icing, in a bowl stir together the powdered sugar and the 1 to 2 tablespoons lime juice until well combined and of drizzling consistency. Drizzle or spread icing on cooled cookies. Sprinkle with additional lime peel.