In large bowl, combine 1 cup butter, ice cream and 1 1/2 teaspoons lemon peel; blend well. Lightly spoon flour into measuring cup; level off. Add flour; mix well. Cover with plastic wrap; refrigerate 1 hour for easier handling.
Heat oven to 375°F. On lightly floured surface, roll dough to 1/8-inch thickness. Cut with floured 1/2-inch round cookie cutter. Coat both sides of cut-out cookies with sugar; place 1 inch apart on ungreased cookie sheets. With fork, prick each cookie 2 to 4 times.
Bake at 375°F. for 7 to 8 minutes or until slightly puffed but not brown. Cool 1 minute; remove from cookie sheets. Cool 10 minutes or until completely cooled.
In small bowl, combine powdered sugar, 1/4 cup butter, 1 teaspoon lemon peel and enough lemon juice for desired spreading consistency; beat at medium speed until smooth. Add food color; mix until well blended. Spread scant 1/2 teaspoon filling between 2 cooled cookies.