Place a pizza stone on a rack in the lower third of an oven. Heat oven to 500 degrees.
On a heavily floured surface, roll out half the dough into a 10-inch circle as thin as possible. Rub with 2 teaspoons olive oil. Using a pizza peel, transfer the dough to the pizza stone, and bake until the dough puffs and is slightly golden brown, about 2 minutes per side. Repeat the process with remaining dough.
Using a towel, remove the crust from the oven, and place on a cutting board. Top half the crust with fennel, zest, 2 teaspoons olive oil, salt and pepper, lemon juice, 10 basil leaves, and mozzarella. Sprinkle the capers, remaining 5 basil leaves, and 2 teaspoons olive oil over the mozzarella. Season with salt and pepper and a final squeeze on the lemon half. Fold the crust in half. Using the back of a heavy saute pan, press the panini together. Repeat process with remaining panini.