Stuffed Zucchini with Pecorino Sauce

Name: Stuffed Zucchini with Pecorino Sauce
Ingredients: 12 firm, very fresh medium zucchini (about 5 1/2 pounds total); 2 tablespoons unsalted butter; 2 tablespoons extra-virgin olive oil; 2 large shallots, finely chopped; 1 tablespoon finely chopped flat-leaf parsley; 1 tablespoon thyme leaves; 1/4 cup freshly grated Parmigiano-Reggiano cheese; Salt and freshly ground pepper; 2 eggs, lightly beaten; 2 hard-cooked large egg yolks; 1/3 cup extra-virgin olive oil; 2 1/2 cups finely grated young sheep's milk cheese, preferably Pecorino Toscano (3/4 pound); 1/3 cup water, plus more if needed; Salt