Heat EVOO over medium heat in a large skillet with a tight-fitting lid. Add garlic and chilis, and stir 2-3 minutes until garlic is very light golden in color. Add tomatoes and season with salt and pepper; cover pan and raise heat to medium high. Cook 10-12 minutes until tomatoes burst, shaking pan occasionally. Uncover and add vinegar, reduce heat to low, reduce a minute or 2 more.
Once tomatoes are in saucepan and have been cooking 5 minutes or so, salt boiling water, drop pasta and cook to al dente. Reserve about 1 cup of starchy cooking water just before draining.
Break tomatoes up a bit with potato masher, add 1/2 cup starchy cooking water and parsley to pan, toss with drained spaghetti for 1-2 minutes using a little more starchy water to combine sauce with pasta if necessary. Serve in shallow bowls topped with Pecorino cheese and basil.
Leftovers are delicious as cold salad as well. Toss with arugula and a little extra oil and vinegar.