Line a baking sheet with parchment paper.

In the work bowl of a food processor, place cold butter, cheeses, thyme, salt, dry mustard, paprika, and pepper; pulse until small clumps form. Add flour, and pulse until combined and large clumps form. (Alternatively, in a large bowl, using a pastry cutter, cut cold butter until butter is pea-size. Add cheeses, thyme, salt, dry mustard, paprika, and pepper. Using a pastry cutter, cut butter mixture until small clumps form. Cut in flour until large clumps form. Knead mixture by hand in bowl or on counter until dough comes together.)

Turn out dough onto a clean surface, and knead until dough comes together. On a lightly floured surface, roll dough to ¼-inch thickness. Using a 1¾-inch round cutter dipped in flour, cut dough, rerolling scraps as necessary, and place on prepared pan. Cover and refrigerate until firm, about 1 hour. (Alternatively, freeze for 15 to 20 minutes.)

Preheat oven to 325°F (170°C). Line 2 baking sheets with parchment paper.

Place chilled dough at least 1 inch apart on prepared pans.

Bake until bottoms are light golden brown, 14 to 16 minutes. Let cool on pans for 5 minutes. Remove from pans, and let cool completely on wire racks.