Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper.
Mix buckwheat flour, Parmigiano-Reggiano cheese, gluten-free flour, and 1 1/2 tablespoon butter in a bowl with your hands until the mixture resembles small pebbles. Mix in sesame seeds and thyme. Add water, 1 tablespoon at a time, until the mixture pulls together into a soft dough.
Divide dough into 6 portions. Let dough rest for 5 minutes.
Roll each portion of dough into a long, thin stick about 6 to 7 inches long. Arrange side by side on the baking sheet. Brush remaining 1 1/2 teaspoon butter over dough.
Bake in the preheated oven until completely browned, about 20 minutes. Cool completely before serving, about 10 minutes.
