Bring a large pot of water to a boil and season generously with kosher salt.

Meanwhile, begin preparing the marinara. Pour the tomatoes and purée into a bowl and crush with your hands or the the back of a spoon, then set aside.

In a large cold skillet, add the extra-virgin olive oil and garlic cloves, and bring to medium heat. As the garlic begins to sizzle, stir occasionally, and cook until fragrant and golden, about 5 minutes. Monitor the heat so the garlic doesn't cook too quickly or burn.

Very carefully stir the tomato paste into the garlic and oil—take care so the oil doesn't splatter and burn you as the paste is added—and cook for 1-2 minutes. Carefully spoon the tomatoes into the pan with the oil and paste and season with salt and pepper. Add the basil sprigs and bring to a boil. Reduce to a bubbling simmer over low heat, stirring occasionally, for 30 minutes. Stir to mix the oil that rises to the top, then discard the basil sprigs and garlic.

Meanwhile, cook the pasta in the boiling water until 2 minutes shy of suggested cooking time on the packaging. Use a slotted spoon or spider to transfer the pasta directly to the sauce to continue cooking until tender but still toothy. Toss the pasta to coat it well, and add 1-2 tablespoons of the pasta water if needed to extend the sauce. Garnish with fresh basil and serve.