Butterfly each of the chicken breasts, making a total of 4 thin chicken breasts. Then flatten with a mallet so that they are uniform in thickness, about 1/4".
Put the pretzels in a food processor and grind until fine. Put the crumbs in a shallow dish. Combine with paprika and season with salt and pepper. In another shallow dish, beat the eggs with a splash of water.
Preheat a large non-stick skillet with a 1/4-inch of vegetable oil over medium-high heat.
One at a time, coat each chicken breast in the egg and then the ground pretzels.
Add the pretzel-coated chicken breasts to the hot oil. Cook in a single layer, in 2 batches if necessary, about 3 or 4 minutes on each side, until cooked through.
Meanwhile, make the sauce. In a saucepan, melt the butter. Whisk in the flour and cook for 1 minute. Whisk in the milk and cook until slightly thickened, 3-5 minutes. Stir in the cheese, mustard, paprika and cayenne. Season with salt and pepper.
Serve chicken breasts with a drizzle of cheddar-mustard sauce and a sprinkle of parsley and red onion.