Prepare the pasta water: Fill a large pot with water and bring it to boil. Add 1 tablespoon of rock salt and allow it to dissolve.
Cook the pasta: Add the Mezzi Paccheri pasta and cook for about 10 minutes or according to the packet's instructions. Stir occasionally to prevent sticking.
Start the sauce: Heat a large pan over medium heat and add 6 tablespoons of EVOO. Add a splash of water and the crushed garlic, then stir. Add the chopped chili and continue stirring for 30 seconds.
Add tomatoes to the sauce: Reduce the heat and add the halved cherry tomatoes. Press down on the tomatoes to release their juice. Increase the heat to medium-high and add the canned cherry tomatoes, pressing them down and mixing well.
Season the sauce: Season with salt and pepper to taste, then tear half of the basil leaves by hand and add them to the sauce. Save the other half for garnishing.
Combine pasta and sauce: Once the pasta is almost ready, scoop up a mug full of pasta water and set it aside. Drain the pasta and add it to the sauce, mixing well. Use the reserved pasta water as needed to adjust the sauce consistency.
Final touches: Add the grated pecorino and Parmigiano Reggiano cheeses, tossing the pasta until the sauce becomes creamy. Garnish with the remaining basil leaves before serving.