Beat butter and sugar in the work bowl of a stand mixer until mixture is creamy; beat egg, egg yolk, lemon zest, lavender, and vanilla extract into butter-sugar mixture at low speed. Beat flour into moist ingredients, 1 cup at a time, to make a workable dough. Form dough into a ball, cover with plastic wrap, and refrigerate for 3 hours.
Preheat oven to 375 degrees F (190 degrees C).
Roll dough out on a floured work surface to a thickness of about 1/8 inch. Cut dough into shapes using cookie cutters and place cookies onto baking sheets.
Bake in the preheated oven until firm, 9 to 10 minutes. Allow cookies to cool on sheets for 2 to 5 minutes before transferring to wire racks to finish cooling.