Preheat oven to 325 degrees. Coat a 4 14-by-13 1/2-inch fluted tart pan with a removable bottom with cooking spray. Line bottom with parchment.
Pulse 1 3/4 cups pretzels in a food processor until powdery. Transfer 1/2 cup to a bowl; discard any remaining pretzel powder. Stir in flour and baking powder. Transfer remaining 1 1/2 cups pretzels to food processor, and pulse a few times until coarsely chopped.
Beat butter and sugar with a mixer on medium speed until pale and fluffy, about 5 minutes. Reduce speed to low, and beat in egg yolk, then pretzel-flour mixture, until combined. Mix in 1/2 cup chopped pretzels.
Press dough evenly into pan. Top dough with 1/4 cup chopped pretzels, pressing slightly; discard any remaining pretzels.
Bake shortbread until pale brown and firm in the center, 25 to 30 minutes. Transfer pan to a wire rack, and let shortbread cool for 1 hour. Remove shortbread from pan, and cut into 1/2-inch-wide rectangles using a serrated knife.
Shortbread can be stored at room temperature for up to 1 week.