Core out peppers by cutting a circle, at an inward angle, around the stem. Remove tops and scoop seeds and membranes out, like a tiny pumpkin.
Cut thumbnail-size cubes of mozzarella, varying size a little bit to fit each pepper. Cut prosciutto slices in half. Wrap each half around mozzarella cubes and stuff into the open peppers.
Pour olive oil into a jar or lidded container. Stir in Parmesan cheese, garlic, oregano, thyme, salt, black pepper, and red pepper. Whisk together well.
Add peppers to the oil mixture and seal the lid. Let marinate for 5 to 6 hours. Serve straight out of the jar.