Fava Bean Pasta e Fagioli

Name: Fava Bean Pasta e Fagioli
Ingredients: 3 pounds fava bean pods, shelled; 1 tablespoon extra-virgin olive oil, plus more for serving; 2 thick slices of lean, meaty bacon, cut crosswise 1/2 inch thick; 1 large carrot, halved lengthwise and cut crosswise into 1/4-inch-thick half-moons; 1 medium onion, cut into 1/2-inch pieces; 1 large garlic clove, very finely chopped; One 3-inch-long rosemary sprig; 1 quart chicken stock or low-sodium broth; 1 cup tubetti or other small pasta; Salt; Freshly ground pepper; Freshly grated Parmigiano-Reggiano cheese, for serving