Make the bechamel: Combine the parmesan, parsley, granulated garlic, nutmeg, 1 1/2 teaspoons salt and 1/2 teaspoon pepper in a small bowl. Melt the butter in a medium saucepan over medium heat; whisk in the flour until a thick paste forms, 1 to 2 minutes. Slowly pour in the heavy cream, whisking constantly. Bring to a simmer and cook, whisking, until thickened, about 1 minute. Whisk in the parmesan mixture until combined. Slowly drizzle in the beaten egg, whisking constantly so it doesn't scramble.
Make the burgers: Heat the olive oil in a large skillet over high heat. Add the beef, granulated garlic and 1/2 teaspoon each salt and pepper. Cook, breaking up the meat with a wooden spoon, until browned, 4 to 5 minutes. Let cool, then drain the meat through a fine-mesh strainer and transfer to a large bowl; set aside.
Line a baking sheet with parchment paper. Add the bechamel to the bowl with the beef; stir to combine. Spread the mixture on the prepared baking sheet and freeze until firm enough to form into patties, 1 to 2 hours.
Put the flour in a shallow dish. Lightly beat the eggs in another shallow dish; spread the breadcrumbs in a third dish. Form the beef mixture into six 3/4-inch-thick patties. Dredge the patties in the flour, then dip in the beaten eggs and coat with the breadcrumbs. Return to the baking sheet and freeze until the coating sets, about 1 hour.
Heat 1 inch of vegetable oil in a large skillet over medium heat until a deep-fry thermometer registers 365 degrees F. Working in 2 batches, add the burgers and fry until golden brown, about 3 minutes per side. Transfer to a paper towel-lined plate.
Top each patty with a slice of cheese while still warm; let sit until the cheese melts, about 5 minutes. Serve the burgers on the buns; top with pickles, mustard and mayonnaise.