Preheat the oven to 425°F. Toss the cherry tomatoes with the minced garlic, 2 tablespoons olive oil, salt, and pepper. Roast for 20 minutes, tossing every 5 minutes to prevent burning.
While the tomatoes are roasting, cook the spaghetti according to the package directions in well-salted water until al dente. Before draining, reserve 1/2 cup of the pasta water.
In a large skillet over medium-low heat, add the remaining 2 tablespoons olive oil, lemon juice, tomato paste, red pepper flakes, salt, and pepper. Stir to combine, then pour in 1/4 cup of the reserved pasta water. Cook for 1–2 minutes.
Add the roasted tomatoes along with all of the garlic, juices, and olive oil from the roasting pan. Cook for 1 minute.
Add the cooked spaghetti and toss until completely coated in the roasted tomato sauce. Add additional reserved pasta water as needed to loosen the sauce.
Divide into bowls and top generously with burrata and fresh basil. Serve immediately and enjoy!