Combine chicken broth and beef broth in bowl; sprinkle gelatin over top and set aside. Pulse onion, carrot, and celery in food processor until finely chopped, about 10 pulses, scraping down bowl as needed; transfer to separate bowl. Pulse pancetta and mortadella in now-empty food processor until finely chopped, about 25 pulses, scraping down bowl as needed; transfer to second bowl. Process chicken livers in now-empty food processor until pureed, about 5 seconds; transfer to third bowl.
Heat oil in large Dutch oven over medium-high heat until shimmering. Add beef, veal, and pork; cook, breaking up pieces with spoon, until all liquid has evaporated and meat begins to sizzle, 10 to 15 minutes. Add chopped pancetta mixture and sage; cook, stirring frequently, until pancetta is translucent, 5 to 7 minutes, adjusting heat to keep fond from burning. Add chopped vegetables and cook, stirring frequently, until softened, 5 to 7 minutes. Add tomato paste and cook, stirring constantly, until rust-colored and fragrant, about 3 minutes.
Stir in wine, scraping pan with wooden spoon to loosen fond. Simmer until sauce has thickened, about 5 minutes. Stir in broth mixture and return to simmer. Reduce heat to low and cook at bare simmer until thickened (wooden spoon should leave trail when dragged through sauce), about 1½ hours.
Stir in pureed chicken livers, bring to boil, and remove from heat. Season with salt and pepper to taste; cover and keep warm.
Bring 4 quarts water to boil in large pot. Add pasta and 1 tablespoon salt and cook, stirring occasionally, until al dente. Reserve ¾ cup cooking water, then drain pasta and return it to pot. Add half of sauce and cooking water to pasta and toss to combine. Transfer to serving bowl and serve, passing cheese separately.
TECHNIQUE – BUILDING A MEATY, SATINY-TEXTURED BOLOGNESE: 1. COOK the ground meats; add depth by sautéing the chopped mortadella, pancetta, and sage in the rendered fat. 2. ADD the soffritto and sweat it until softened and then add concentrated tomato flavor in the form of tomato paste. 3. DEGLAZE the pot with wine; stir in the broth plus the bloomed gelatin to develop luxurious silky texture. 4. STIR in the pureed chicken livers for subtle but rich taste.