Stir olive oil and garlic together in a small bowl until combined. Let sit at room temperature until flavors are infused, about 3 hours.

Preheat the oven to 300 degrees F (150 degrees C). Line a baking sheet with a silicone baking mat.

Place bread cubes in a large bowl. Pour garlic oil through a strainer onto bread; toss until bread is evenly coated with oil. Add 1/2 cup Parmigiano-Reggiano cheese, Italian seasoning, paprika, black pepper, salt, and cayenne pepper; toss to coat. Spread bread cubes out in a single layer on the prepared baking sheet.

Bake in the preheated oven for 15 minutes. Turn croutons and bake for 15 minutes more. Push croutons together with a spatula to form a compact rectangle on the baking sheet. Sprinkle remaining 1/4 cup Parmigiano-Reggiano cheese over the top. Continue baking until cheese is melted and croutons are crunchy, 10 to 15 minutes more.