In a large bowl, combine beef, garlic, parsley, Parmesan, onion, bread crumbs, egg, tomato paste, salt, pepper, and oregano with your hands. Form the mixture into tablespoon size balls and dredge in the flour.
Preheat the oven to 375ºF.
In a large skillet heat olive oil over medium heat. Brown the meatballs on all sides, working in batches and adding more olive oil as necessary. Place browned meatballs on a baking sheet and bake in the oven for 10 minutes.
In a large saucepan, heat spaghetti sauce over medium low heat. Add meatballs and stir gently to coat in the sauce. Bring to a bubble and reduce heat to low. Simmer anywhere from 15 minutes to several hours, gently stirring occasionally until you're ready to assemble the sandwiches.
To make the sandwiches, preheat the oven to 375ºF. Slice hoagies in half and place 3-4 meatballs inside. Cover with a slice of cheese and place on a baking sheet. Place in the oven until the cheese is melted and the bread is toasty, about 5 minutes. Serve immediately.