Line a 9x9 pan with parchment paper; set aside. In a saucepan over medium heat, melt butter. Add flour and cook for 1:30 to 2 minutes, stirring constantly.
Pour in milk, a little at a time, and whisk until blended before adding more.
By the handful, add cheeses and stir until melted between each addition. When there’s a little cheese left, add the rest of it along with egg yolk, smoked paprika, and freshly ground mixed peppercorns; mixed until well incorporated. Spoon the cheese mixture into the prepared pan and spread it out with a spatula.
Place a piece of wax paper, making sure it touches the surface, and level it out using a flat bottom measuring cup or glass.
Transfer to the fridge for at least 6 hours.
In a shallow skillet, bring the oil temperature to 375ºF and prepare the breading station into 3 shallow dishes: one with flour, the second one with egg and egg white, beaten together, and the last one with breadcrumbs. Remove the cheese mixture from the fridge and lift it out from the pan; peel off the wax paper and discard it.
Cut the mixture into 9 equal pieces.
Working with one at a time, dredge a piece in flour, coating all sides, and gently shake it to remove any excess. Dip into the egg mixture, covering the piece entirely, and hold it to allow the eggs to drip off. Lastly, dredge in breadcrumbs, covering the piece completely; transfer to a plate or cutting board while working with the other ones.
Without crowding the skillet and working in batches, fry a few breaded pieces until golden brown, about 1 minute per side.
Transfer to a plate lined with paper towels to absorb any excess oil and set them aside while frying the other pieces.