Preheat the oven to 325 degrees F. Line 2 baking sheets with parchment paper.
Place the butter and 1/2 cup of the confectioners' sugar in a stand mixer fitted with a paddle attachment. Beat until fluffy, about 2 minutes.
Meanwhile, place the walnuts in the carafe of a food processor. Pulse until almost finely ground, checking after every 5 seconds, about 30 seconds total.
Whisk together the ground walnuts, flour, cornstarch, cayenne and salt in a large bowl. Pour half of the dry ingredients into the butter mixture and beat on low speed until combined. Repeat with the remaining dry ingredients until a thick dough comes together.
Scoop the dough by heaping tablespoons and roll into a smooth ball in between your palms. Place onto the prepared baking sheets, leaving about 1 inch of space between each cookie.
Bake until golden brown, 16 to 18 minutes. Allow to cool on the baking sheets for 3 minutes before moving to a wire rack to cool completely.
Place the remaining 1 cup confectioners' sugar in a large bowl. Working in batches, add the cooled cookies to the sugar and toss until evenly coated in sugar. Sprinkle the tops of the cookies lightly with paprika if desired before serving.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)