Position oven racks in the upper and lower thirds of the oven and preheat to 375 degrees F. Line 2 rimmed baking sheets with parchment paper.
Combine the flour, salt, cayenne, paprika, garlic powder and a few grinds of black pepper in a food processor and pulse until mixed. Add the Cheddar, butter and sour cream and process until the mixture forms a ball. Form the dough into a disc, then wrap it in plastic wrap and refrigerate until firm, about 30 minutes.
Break off a tablespoon of dough and roll it between your hands or on a work surface to form a rope about 10 inches long. Twist the rope into a pretzel shape and place it on one of the prepared baking sheets. Repeat with the remaining dough to make 24 twists (12 per baking sheet).
Beat the egg with a small splash of water. Brush the twists very lightly with the egg wash. Put a little paprika in a small sieve and dust over the twists.
Bake the twists, switching the positions of the baking sheets halfway through, until golden brown on the underside and pale golden on top, about 12 minutes.