Place chicken cutlets between plastic wrap and pound to an even 0.5-inch thickness.
Season chicken with 0.5 teaspoon salt and 0.25 teaspoon black pepper.
Prepare three shallow bowls: one with 0.5 cup flour, one with beaten eggs, and one with Panko mixed with garlic powder and smoked paprika.
Dredge each cutlet in flour, shaking off excess, then dip in egg, and finally coat thoroughly in Panko mixture.
Heat vegetable oil in a large skillet over medium-high heat until shimmering.
Fry chicken for 4 to 5 minutes per side until golden brown and the internal temperature reaches 165 degrees Fahrenheit. Transfer to a wire rack to rest.
In a small saucepan over medium heat, melt butter. Whisk in 2 tablespoons flour and cook for 1 minute without browning.
Slowly whisk in milk and dry mustard. Continue whisking until the sauce thickens and reaches a light simmer.
Reduce heat to low and stir in shredded cheddar cheese until completely melted and smooth. Season with remaining salt and pepper.
Serve crispy chicken immediately with the warm cheddar sauce ladled over the top.