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Place stock in a 1-cup liquid measure and sprinkle with gelatin. Set aside.
Heat olive oil in a pressure cooker over medium-high heat until shimmering. Add pancetta and cook, stirring frequently, until pancetta is browned and crisp, about 12 minutes. Add onions, carrots, celery, garlic, sage, and half of parsley and cook, stirring, until softened but not browned, about 8 minutes.
Increase heat to high, add chicken livers, and cook, stirring, until livers are no longer pink, about 5 minutes. Add beef and pork, season with salt and pepper, and cook, stirring and breaking up meat with a wooden spoon or a potato masher, until meat is no longer pink, about 10 minutes. Continue cooking, stirring occasionally, until excess liquid has evaporated and the meat starts to sizzle, about 25 minutes.
Add stock and gelatin mixture, wine, tomatoes, 1 cup heavy cream, and bay leaves. Seal and cook at high pressure (12 to 15 psi) for 30 minutes. Release pressure and remove lid. Simmer over moderate heat until thick and emulsified, 30 to 45 minutes longer.
Stir in remaining 1/2 cup heavy cream, Parmesan, fish sauce, basil, and remaining parsley. Bring to a boil, stirring constantly to emulsify. Season to taste with salt and pepper. Bolognese can be cooled and stored in sealed containers in the refrigerator for up to 1 week.
To Serve: Heat Bolognese in a large pot until just simmering. Set aside. Cook pasta in a large pot of well-salted water until just barely al dente. Drain, reserving 1/2 cup of cooking liquid. Transfer to a large skillet or sauteuse and add 3/4 of sauce, along with cooking water. Cook over high heat, tossing and stirring gently, until sauce is thick and pasta is coated, about 30 seconds. Transfer to a serving bowl and top with remaining sauce. Serve immediately, passing extra Parmesan at the table.