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Preheat oven to 425°F. Whisk together flour, cornstarch, baking powder, salt, garlic powder, and baking soda in a large bowl. Cut in butter using a pastry blender or 2 forks until mixture is crumbly and butter is pea-sized. Stir in cheese, pickles, and dill until evenly dispersed.
Whisk together sour cream, milk, and pickle brine in a small bowl; stir into flour mixture just until combined. Press and fold flour mixture together by hand until a rough, shaggy dough forms
Turn dough out onto a very lightly floured work surface. Pat and shape dough into a 12- x 5 ½-inch rectangle. Fold dough rectangle into thirds like a letter; pat dough back into a 12- x 5 ½-inch rectangle, very lightly flouring work surface and top of dough as needed. Repeat folding and patting procedure twice; roll dough to a ¾-inch thickness.
Cut dough using a floured 2 ½-inch round cutter, rerolling and cutting scraps as needed. Place dough rounds at least ½ inch apart on a parchment paper–lined baking sheet. Chill in freezer, uncovered, until slightly firmed, about 15 minutes.
Brush tops of chilled dough with beaten egg. Bake until golden and cooked through, 20 to 25 minutes. Let stand on baking sheet on a wire rack 10 minutes. Serve warm.