In a large bowl, beat the 1 cup butter with an electric mixer for 30 seconds or until softened. Add 1/2 cup powdered sugar and lemon flavoring; beat well. Add flour and salt. Beat until well mixed. (If necessary, stir in last part of flour.)
Shape into 1-inch balls and place 2 inches apart on ungreased cookie sheets. Flatten to 2-inch circles with the bottom of a glass dipped in granulated sugar. Bake in 375 degree F oven for 8 to 10 minutes or until edges are lightly browned. Cool cookies on wire rack.
For filling, in a small saucepan, stir together 2/3 cup granulated sugar and cornstarch. Stir in lemon peel and juice. Cook and stir over medium heat until mixture is thickened and bubbly.
Cook and stir for 2 minutes more. Gradually stir about half of the hot mixture into beaten egg. Return all of the egg mixture to saucepan. Cook and stir for 2 minutes more; do not boil. Remove from heat.
Stir in 1 tablespoon butter. Cover surface with plastic wrap; chill.
To assemble cookies, spread about 1-1/2 teaspoons filling on flat sides of half the cookies; top with remaining cookies, flat sides toward filling. Store filled cookies in the refrigerator. Makes about 24 sandwich cookies.
To spruce up these treats, you can dip the glass in colored sugar before flattening the cookie dough. Roll the baked and filled cookies in powdered sugar, if you like.