In a mixing bowl, whisk together the flour, corn starch, baking powder, baking soda, salt, and garlic powder. Cut the butter into it and mix well until it's crumbly or sandy, with no pieces larger than about pea-sized.
Mix the shredded cheese, pickle, and dill into the flour mixture.
In another bowl, mix the sour cream, milk, and pickle juice together. Then, add this mix to the flour mixture.
Stir this all together until it just combines. Using your hands, start pressing and folding the dough to keep mixing until it's rough and shaggy.
Prepare a lightly floured surface and turn the dough out onto it. Shape the dough into a rough rectangle, about 12" by 5", then fold it in thirds like a letter. Do this twice more, adding more flour as necessary if it sticks to the surface too much.
Roll the dough out to a final thickness of about ¾". Using a 2½"-3" biscuit or cookie cutter, cut rounds out of the dough. Arrange these onto your baking sheet and chill them in the freezer for around 10-15 minutes to firm them up.
Preheat your oven to 425℉. Remove the biscuits from the freezer and brush the tops with beaten egg.
Bake for 20-25 minutes, or until they're cooked through and golden. Be aware that a bunch of butter will melt and leak out, and the interior might seem overly moist and under-done because of it. Don't worry; they're still fine.
When done, place the biscuits on a wire rack to cool and enjoy when warm.