In a food processor, combine the flour, baking powder, salt, and cayenne and pulse to combine.
Add the butter and cheese and pulse until well mixed.
Add 4 tablespoons of ice water and pulse until the mixture starts to hold together. Add additional water if necessary.
Divide the dough in half, form into circles and wrap in plastic wrap. Freeze 20 minutes.
Line two baking sheets with silicone mats or parchment. Using one square at a time, roll the dough on a lightly floured surface to a scant 1/8 inch thickness.
Using a fluted 1 3/4-inch cookie cutter, cut the dough into circles, rerolling the dough to use it up.
Poke holes in each with a fork and freeze until firm, 10 to 15 minutes. Repeat with the remaining dough.
Preheat the oven to 350°F.
Brush the circles with the egg and sprinkle with salt.
Bake until puffed and crisp, 18 to 21 minutes. Transfer to a cooling rack and repeat with the remaining dough.
Store in airtight containers at room temperature.
