To prepare sauce, heat 1 teaspoon oil in a large Dutch oven over medium-high heat. Add beef and pork; cook 3 minutes, browning on all sides. Remove meat from pan, and cool slightly. Finely chop meat.
Heat pan over medium heat. Add onion, celery, carrot, porcini, and garlic cloves; cook 10 minutes, stirring occasionally. Add chopped meat, milk, 3/4 teaspoon salt, tomatoes, and bay leaf; bring to a boil. Cover, reduce heat, and simmer 1 hour. Uncover and cook 45 minutes, stirring occasionally. Stir in parsley and thyme; cook 15 minutes or until slightly thick, stirring occasionally. Discard bay leaf.
To prepare the gnocchi, preheat oven to 400°.
Bake potatoes at 400° for 1 1/2 hours or until done; cool slightly. Cut each potato in half lengthwise; scoop out pulp. Discard skins. Mash pulp. Place 4 cups mashed potatoes in a large bowl, reserving remaining mashed potatoes for another use. Lightly spoon flour into dry measuring cups; level with a knife. Add 1 cup flour, 1/4 teaspoon salt, and egg yolks to 4 cups mashed potatoes, and stir to combine. Knead until smooth (about 2 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel tacky).
Divide dough into 6 portions. Shape each portion into a 10-inch-long rope. Cut each rope into 10 (1-inch) pieces; roll each piece into a ball. Working with one dough piece at a time (cover remaining dough to prevent drying), using your thumb or index finger, roll dough piece down the tines of a lightly floured fork (gnocchi will have ridges on one side and an indention on the other). Place gnocchi on a lightly floured baking sheet.
Bring 1 gallon water to a boil in a large stockpot. Add half of gnocchi; cook 3 minutes or until done (gnocchi will rise to the surface). Remove cooked gnocchi with a slotted spoon; place in a colander. Repeat procedure with remaining uncooked gnocchi. Serve immediately with sauce. Sprinkle with cheese.