Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper or a silicone baking mat.
Unfold puff pastry into a flat sheet and brush olive oil over top. Season with salt, pepper, and cayenne pepper.
Sprinkle Cheddar and 1/4 cup Parmigiano-Reggiano all over dough to cover. Cover with a piece of plastic wrap. Press on the plastic wrap with your fingers to push cheese and seasonings firmly into the dough, or place a sheet pan over top and press down. Remove and discard plastic wrap.
Cut dough along the seams into thirds with a pizza cutter or sharp knife, then cut each third lengthwise into thirds again for a total of nine strips.
Pick up one strip and place it seasoned-side down onto the work surface. Twist it from both ends eight to nine times to make a rolled tube of dough with the seasoning facing out. Place onto the prepared baking sheet and repeat with remaining strips.
Sprinkle remaining 1 tablespoon Parmigiano-Reggiano cheese over cheese sticks. Roll sticks lightly to even up their shapes; pick up and press any dropped cheese onto the surfaces.
Bake in the preheated oven for about 10 minutes, then flip and bake until sticks are browned and crisp and hold their shape without drooping when lifted, 10 to 20 more minutes. Cool on a wire rack before serving.