Combine flour, xanthan gum, brown sugar, and salt in a medium bowl.
Add olive oil and stir until a dough forms. Stir in thyme leaves.
Roll out dough between two pieces of parchment paper. Freeze for 30 minutes.
Preheat oven to 325°.
Cut cookies out with a 2 1/2-inch cutter. Place on a baking sheet lined with parchment paper. Gather scraps; reroll and cut to use all of dough.
Bake at 325° for 28 minutes or until fragrant and golden. Allow to cool.