Trim away any greenish spots and bits of fat from the livers. Rinse in cold water, cut each liver into 3 or 4 pieces, and pat very dry with paper towels.
Put the shallot or onion in a small saucepan or saute pan with the oil and butter over medium heat. Cook and stir until translucent. Add the chopped garlic and cook briefly, without letting it color.
Add pancetta or prosciutto and the sage. Stir and cook about a minute or less. Add the ground beef with a large pinch of salt and a few grindings of pepper, then crumble and cook until the beef loses its raw red color.
Add the chicken livers, raise heat to medium high, stir well, and cook briefly, just until the livers lose their raw red color.
Add the tomato paste and vermouth mixture and cook 5 to 8 minutes, stirring from time to time. Taste and adjust salt.
Turn the sauce into a bowl with cooked, drained pasta and toss well. Serve at once with grated Parmigiano-Reggiano.
