Cook spaghetti in 1 tablespoon of kosher salt-seasoned boiling water for 5 minutes. Save 2 cups of pasta cooking liquid for the sauce. While the pasta cooks, start the sauce.
Set aside the sliced tomatoes for the sauce. Heat 3 tablespoons olive oil and 3 garlic cloves in a large sauté pan over medium-high heat. Tilt the pan so the garlic cooks in the oil.
Carefully add the tomatoes and juice to the seasoned oil when the garlic becomes aromatic and browns. Cook tomatoes for 1-2 minutes, breaking them up.
Discard the garlic. Cook the sauce for 6-8 minutes, stirring periodically, until creamy. When the sauce is almost done, add the basil.
While the pasta boils, heat the remaining olive oil and garlic in a sauté pan. Cook garlic until golden, then discard. 1/4 cup at a time, add pasta water to garlic-infused oil.
Return the noodles to the pan and cook until the water and oil are absorbed. Sauce four plates with noodles.
Garnish with basil and olive oil before serving. Serve noodles in sauce family-style.