Preheat oven to 300. Line a baking sheet with a silicone baking mat.
Stir olive oil and garlic together in a bowl. Let sit at room temperature until flavors are infused, about 3 hours.
Place bread cubes in a large bowl. Pour garlic oil through a strainer onto the bread; toss to coat bread 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 15 minutes more. Push croutons together with a spatula so that they are in a compact rectangle on the baking sheet and no longer spread out; sprinkle remaining 1/4 cup cheese over the top. Bake until cheese is melted and the bread is crispy and crunchy, 10 to 15 minutes more.