Char the onion halves cut-side down in a large stockpot over medium heat until deeply browned, 5–7 minutes.
Add the carrots, celery, chicken pieces, beef shank, and water. Bring just to a simmer over medium heat, then skim off any foam.
Reduce to low and simmer gently, uncovered or partially covered, until the broth is full-flavored and clear, 3 hours, skimming occasionally; keep the surface at a bare tremble, not a boil.
Strain the broth through a fine-mesh sieve into a clean pot; discard solids. Return the clear broth to a gentle simmer and season with kosher salt to taste. Keep hot.
Make the dough: Mound the 00 flour on a clean work surface and make a wide well. Crack in the eggs and beat with a fork, gradually drawing in flour until a shaggy dough forms. Knead until smooth and elastic, 8–10 minutes. Wrap and rest at room temperature for 30 minutes.
Make the filling: In a food processor, add the cooled pork loin, prosciutto, mortadella, Parmigiano Reggiano cheese, nutmeg, black pepper, and fine sea salt. Pulse to a very fine, cohesive mixture without turning it pasty, scraping the bowl as needed. To check seasoning, fry a pea-sized bit in a skillet and adjust salt or pepper if needed.
Roll the pasta: Cut the rested dough into 2 pieces. Working with one piece at a time (keep the other wrapped), roll with a machine or pin to a very thin sheet about 1/32 inch (0.8 mm). Lay on a lightly floured surface and cover with a towel.
Cut and fill: Using a knife or pastry wheel, cut the sheet into 1.25-inch (3 cm) squares. Place a pea-sized mound (about 1/4 teaspoon) of filling in the center of each square.
Shape the tortellini: Fold each square into a triangle, pressing out any air and sealing the edges firmly. Wrap the two corners around your fingertip and pinch to seal into a ring. Arrange on a lightly floured tray and cover while you finish the rest.
Cook in brodo: Bring the clear broth back to a gentle boil. Add the tortellini and cook until they float and the pasta is tender with a slight bite, 2–4 minutes.
Serve: Ladle tortellini and brodo into warm bowls. Serve with Parmigiano Reggiano cheese, finely grated (for serving).