Peel the garlic cloves and cut into very thin slices.
In a small saucepan, add the olive oil and the garlic slices and gently cook over very low heat until the garlic is a light golden color (not browned). Remove from heat and set aside.
On a cutting board, cut the heirloom tomatoes in half and scoop out the excess seeds and pulp with a spoon.
Discard the excess seeds and pulp. Chop the tomatoes into a rough 1/2-inch dice and put into a large pasta serving bowl. Season with salt and freshly ground pepper.
Remove the garlic from the seasoned oil with a slotted spoon.
Add the garlic-infused olive oil to the tomato mixture, season with salt and pepper and set aside while the pasta is cooking. In the time the pasta water comes to a boil and the pasta is cooking, the raw tomatoes will become infused with the garlic-scented oil.
Cook pasta according to package directions (usually about 10 minutes for short pasta, but check the label for cooking times).
Remove cooked pasta from cooking water with a slotted spoon and add to the serving bowl with tomato-oil mixture. Continue adding and stirring until the pasta and tomato mixture are incorporated.
Add the burrata cheese to the pasta and tomato mixture and give it a gentle stir.
Gently tear the basil leaves and add to the pasta.
Serve in pasta bowls and add freshly grated Parmigiana cheese to each bowl. Finish with a drizzle of extra virgin olive oil and serve.