Bring 6 quarts of water to a boil and add 2 tablespoons salt.

In a 12 to 14 inch sauté pan, heat 2 tablespoons olive oil over medium heat. Add the onion and garlic and stir over medium heat, stirring occasionally, until the onions and garlic are softened, about 5 minutes. Add the zucchini (without the blossoms) to the pan and cook, over medium-high heat, until the zucchini is tender, about 5 minutes more. Remove from the heat.

Cook the pasta according to the package directions, until tender yet al dente. Return the squash mixture to the heat, drain the pasta, and add it to the pan. Add the chili flakes, parsley, zucchini blossoms, and the remaining 2 tablespoons of olive oil and toss over high heat until the pasta is heated through, about 1 minute. Divide the pasta evenly among 4 warmed pasta bowls, grate the Parmigiano-Reggiano over the top, and serve immediately.