Combine flour, thyme leaves, and salt in a small bowl, and set aside.
Process butter and sugar in a food processor until smooth and well combined, stopping to scrape down sides. Add vanilla; process 15 seconds.
Add flour mixture; pulse 10 to 12 times until dough pulls away from sides of bowl and forms a ball.
Divide dough in half. Roll each half to ¼-inch thickness on a lightly floured surface, adding just enough flour to keep the dough from sticking. Cut with a 2-inch cookie cutter, and place cookies 1 inch apart on parchment-lined cookie sheets. Place cookie sheets in freezer for 20 minutes or until cookies are firm to the touch.
Bake at 275 for 30 minutes or just until edges begin to brown. Remove to wire racks to cool completely.