Preheat oven to 250F.
Add the garlic, tomatoes, extra virgin olive oil, thyme, salt and red pepper flakes into a medium baking dish. Roast the tomatoes for about an hour until the tomatoes develop some char and the garlic become tender. Then remove the baking dish from the oven.
During the last 30 minutes of roasting time, bring a large pot of water to boil. When the water starts boiling, generously salt the water (1 tbsp of kosher salt) and cook the tagliatelle al dente. Drain the pasta and transfer to a large bowl.
Gently smash the garlic and tomatoes with a fork or potato masher. (Note: Be careful when smashing the tomatoes because the hot juices from the tomatoes may splatter on you.) Then drain the tomatoes and garlic of the oil and set them aside.
Transfer the drained tomatoes and garlic to the large bowl of pasta. Then add a 1/4 cup of the tomato and garlic oil and 1/2 cup of the reserved pasta water. Add the grated cheeses and parsley or basil. Mix all the ingredients together . Season with salt and black pepper. Serve immediately. Enjoy!