Bring a large pot of lightly salted water to a rolling boil. Cook angel hair pasta at a boil until tender but still slightly firm, 4 to 5 minutes. Drain the pasta, reserving 1 cup of the cooking water for later use. Transfer the drained pasta to a large serving bowl.
Place tomatoes in a bowl; season with sea salt and black pepper.
Pour 1/3 cup extra-virgin olive oil over the tomato mixture and gently toss; set aside.
Heat 1 tablespoon olive oil in a small skillet over medium heat.
Cook and stir sliced shallot in the hot oil until just becoming translucent, 1 to 2 minutes.
Stir garlic into the shallot; continue cooking and stirring 1 minute more. Remove the skillet from heat.
Assemble the dish by adding the tomato mixture, the shallot and garlic mixture, and the prosciutto to the pasta; toss to combine. Use the reserved water from cooking the pasta to make it easier to mix.
Top the pasta dish with basil and Parmigiano-Reggiano cheese to serve.