Bring a large pot of lightly salted water to a boil. Fill a large bowl with ice and cold water.
Score a shallow "x" in the bottom of the tomatoes. Add tomatoes to the boiling water and cook uncovered until skin starts to peel off, about 1 minute. Drain in a colander and immediately immerse in ice water for several minutes to stop the cooking process. Peel tomatoes.
Transfer peeled tomatoes to a food mill, and process until smooth.
Meanwhile, bring a large pot of lightly salted water to a boil. Cook angel hair pasta in the boiling water, stirring occasionally, until tender yet firm to the bite, 4 to 5 minutes. Drain, saving 1/4 cup of the pasta water for the sauce.
While pasta cooks, melt butter in a large skillet over low heat. Add garlic and salt and cook until fragrant, 1 to 2 minutes. Add pureed tomatoes and as much pasta water as desired. Simmer until thickened, 3 to 5 minutes. Season with salt and pepper to taste.
Divide pasta amongst 4 serving bowls. Top with tomato sauce and sprinkle with fresh basil. Serve immediately.