Preheat oven to 400 degrees F. Add the sliced garlic and grape tomatoes to one half of a baking sheet. Season generously with salt and pepper. Add 1/2 tsp oregano and olive oil and combine everything well.
Lay out the strips of prosciutto to the other side of the baking sheet. Place baking sheet in the preheated oven. Watch closely because the prosciutto may only take 10-12 minutes to get crispy. Remove the prosciutto once it's ready. The tomatoes will take closer to 20 minutes. You want the prosciutto be crispy but not burnt. The tomatoes are ready once they start to burst open.
Meanwhile, cook the rigatoni according to package instructions. Drain and add to a large bowl or back to the pot you cooked the pasta in. When the tomatoes are done roasting, add them straight to the cooked pasta. Add another 1/4 tsp of oregano and red pepper flakes.
Serve immediately. Top each bowl with crispy prosciutto, fresh basil, and burrata. Season the burrata with salt and pepper to taste.