Pound pork chops on a solid, level surface between two sheets of heavy plastic (resealable freezer bags work well) with the smooth side of a meat mallet until 1/8-inch-thick. Trim any fat from edges of chops; make a few small slits along edges to prevent curling during cooking.

Whisk flour, 1 teaspoon salt, and pepper together in a bowl. Whisk milk and egg together in another bowl until combined. Combine panko bread crumbs and paprika together in a third bowl. Dredge pork chops, one at a time, into flour mixture. Dip into egg mixture, letting excess drip back into bowl. Press into panko mixture to coat both sides of chops. Place breaded chops, unstacked, onto a plate. Repeat with remaining chops.

Preheat oven to 200 degrees F (95 degrees C).

Heat vegetable oil in a large skillet over medium-high heat. Brown chops, two at a time, in skillet until golden brown and no longer pink in centers, 3 to 4 minutes per side. Place chops in preheated oven to keep warm.

Add chicken stock to skillet; reduce heat to low, and scrape up any brown bits on bottom of the skillet. As stock simmers, whisk sour cream, dill, and 1/2 teaspoon salt together in a bowl; whisk in about 1/4 cup hot stock then pour back into the skillet. Cook and stir until warm and thick, about 5 minutes; serve with pork chops.