Heat the oven to 350° F with a 10-inch cast-iron skillet in it.
In a small bowl, combine 2 teaspoons each of the Parmesan and pecorino and 1 teaspoon of the pepper. Set aside.
In the bowl of a stand mixer fitted with the paddle, beat the butter on low speed for 1 minute or so, until it’s smooth and fluffy, like cake frosting. Add the sugar, salt, and remaining 1 teaspoon pepper and mix until combined. Turn off the mixer and, using a rubber spatula, scrape down the sides of the bowl. Set the speed to medium and mix for 4 to 5 minutes more, until the mixture takes on a thick, creamy, almost shiny texture, like mayonnaise.
Turn off the mixer and scrape down the sides of the bowl again. Add the all-purpose and semolina flours and mix on low speed to incorporate. Turn off the mixer, scrape down the sides of the bowl one more time, add the remaining 1/2 cup Parmesan and 1/2 cup pecorino, and mix for 1 minute. Using the rubber spatula, push the dough together to form a ball.
Remove the hot skillet from the oven and brush it with 1 teaspoon of the olive oil. Turn the dough into the skillet and, working quickly, using your fingers (but being careful of the hot pan), press the dough into the skillet, pushing it out to fill the edges and flattening it to create an even surface. Brush with the remaining 1 tablespoon plus 1 teaspoon olive oil. Sprinkle the dough with the cheese-pepper mixture.
Bake the shortbread for 18 to 23 minutes, until the edges begin to brown. The middle should be cooked through, but slightly soft; it will harden as it cools. Let cool for 10 minutes. Using a plate, carefully invert the pan and flip the shortbread out, then flip it once more onto another plate, so that it’s right-side up. Let cool completely. (Or simply cool, slice, and serve from the pan.)
To serve, divide the shortbread into 10 to 12 wedges. Enjoy it with your afternoon coffee or aperitifs. Like prosecco, Bellinis, rosé, or whatever you like to drink at cocktail hour.
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