Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain the pasta, reserving 2 cups of the pasta water.

Meanwhile, heat the oil in a large nonstick frying pan over medium heat. Add the shallots, garlic, carrots, salt, and pepper. Cook until tender, about 8 minutes. Add the tomato paste and 1/2 cup of the hot pasta water. Stir to melt the tomato paste and create a sauce, adding more pasta water if necessary. Stir in the oregano, thyme, and parsley. Gently fold in the cooked pasta, peas, and the cheeses, adding more reserved pasta water if necessary. Transfer to a platter and serve immediately.