Preheat the oven to 425 degrees F. Line 2 baking sheets with parchment paper.
In a saucepan, heat the milk, butter, 1 teaspoon salt, and 1 teaspoon pepper over medium heat, just until scalded. Add the flour all at once and beat it vigorously with a wooden spoon until the mixture comes together. Lower the heat and cook, stirring constantly, for 2 minutes. Dump the mixture into the bowl of a food processor fitted with the steel blade. Immediately add the eggs, Pecorino, and Parmesan and pulse until the eggs are incorporated and the dough is smooth and thick.
Spoon the mixture into a pastry bag fitted with a large, plain 5/8-inch round pastry tip. Pipe the dough in mounds 1 1/4 inches wide and 3/4 inch high (1 inch apart) onto the prepared baking sheets. With a wet finger, lightly press down the swirl at the top of each puff. (You can also use two spoons to scoop the mixture and shape the puffs with damp fingers.) Brush the top of each puff lightly with the egg wash, and sprinkle with Pecorino, salt, and pepper. Bake for 15 to 20 minutes, until golden brown outside but still soft inside. Serve hot.