In a medium bowl, mix butter and brown sugar with an electric mixer until combined. Blend in lemon zest, lemon extract, and vanilla.
Stir flour and salt into butter mixture. Transfer mixture to waxed paper or plastic wrap. Roll into a log (for slicing cookies) or shape into a disk (for using cutters); wrap dough and refrigerate for 30 minutes.
Preheat oven to 325 degrees F (170 degrees C).
Roll dough out on a lightly floured surface until 1/4 inch thick. Cut out cookies using a 2-inch round cutter and place 2 inches apart on ungreased baking sheets. (Or slice cookies and arrange on baking sheets). Gather and reroll dough scraps.
Bake in preheated oven until edges are pale golden, but not browned, about 20 to 25 minutes. Let stand on baking sheet 2 minutes. Remove to a rack and let cool completely.