Preheat the panini press according to the manufacturer's instructions.
Arrange the 12 slices of bread on a clean work surface. Spread 1 tablespoon of the pizza sauce on one side of each slice (this uses all 6 ounces).
For each sandwich, place one sauced slice sauce-side up. Evenly divide the 16 ounces sliced mozzarella and 3 ounces pepperoni among the 6 sandwiches, layering the cheese and pepperoni on top of the sauce on each bottom slice.
Close each sandwich with a second slice of bread so the sauce sides are facing inward.
Lightly brush both outside faces of all sandwiches with the 2 tablespoons of olive oil (or melted butter), using the full amount across the batch.
Place sandwiches in the preheated panini press (work in batches if needed) and cook until the bread is golden brown and the cheese is melted—about 3–5 minutes, depending on your press.
Remove the paninis from the press and let them rest 1 minute before slicing and serving.