Prepare the chicken breasts by patting them dry and trimming any excess fat.
In a shallow dish, combine all-purpose flour, garlic powder, onion powder, paprika, salt, and pepper. In another dish, beat the eggs. Place pretzel crumbs in a third dish.
In a medium saucepan, melt butter over medium heat. Whisk in flour and cook for 1 minute. Gradually add milk, whisking continuously until thickened (3-5 minutes). Stir in cheddar cheese and Dijon mustard, season with salt and pepper.
Dredge each chicken breast in seasoned flour, dip in beaten eggs, and press into pretzel crumbs to coat.
Preheat oven to 400°F. Heat vegetable oil in a skillet over medium heat. Cook chicken for 3-4 minutes on each side until golden brown. Transfer to a baking sheet and bake for 15-20 minutes until internal temperature reaches 165°F.