Bring large pot of well salted water to boil and cook pasta according to directions. When draining reserve 1 cup of pasta cooking water.
Meanwhile heat 3 Tbsp olive oil in large skillet over medium heat. Add garlic and tomatoes and cook until tomatoes begin to break down releasing their juices, 5-10 mins. Add red pepper flakes to taste.
Return pasta to pot, add tomatoes mixture and enough pasta cooking water that tomatoes are saucy and coat pasta. Add about 1/2 cup grated parmesan to pasta and toss together. Taste and adjust for salt and pepper.
Serve pasta in large, flat bowls sprinkled with slivered basil, grate more parmesan on top and drizzle with remaing Tbsp olive oil.