In a large saucepan, heat the oil until shimmering. Add the garlic and cook over moderate heat until lightly browned, about 1 minute. Add the carrots, celery and onion and cook, stirring, until softened, about 10 minutes. Stir in the chicken livers and sage and cook, stirring occasionally, just until the livers lose their pink color, about 3 minutes. Add the ground veal, pork and lamb and season with salt and pepper. Break up the meat with a wooden spoon and cook over high heat until no trace of pink remains, about 8 minutes. Pour in the wine and cook over high heat, stirring, until reduced by half, about 15 minutes.
Stir in the stock, tomatoes and basil and season with salt and pepper. Cover partially and simmer the ragù over moderately low heat, stirring occasionally, until the liquid has reduced by half, about 45 minutes. Uncover, add the cream and cook for 10 minutes longer. Remove the ragù from the heat and skim off the fat, using a ladle.
Meanwhile, in a large pot of boiling salted water, cook the lasagna noodles until al dente. Drain and rinse under cold water, then pat dry and spread out on 2 clean kitchen towels.
Preheat the oven to 350°. Spread 1 cup of the meat ragù in a 9-by-13-inch glass baking dish and top with a layer made of 4 noodles. Spread 1 1/3 cups of the ragù over the noodles and dollop with scant tablespoons from 2/3 cup of the White Sauce, then sprinkle with 2 tablespoons of the Parmesan. Repeat to make 4 more layers. Top with the remaining lasagna noodles, White Sauce and Parmesan cheese. Reserve any remaining ragù for passing tableside.
Bake the lasagna for 35 minutes, or until heated through. Preheat the broiler. Broil the lasagna until the top is bubbling and golden, about 1 minute. Let stand for 10 minutes before serving.
The lasagna can be assembled and refrigerated for 1 day. Bring to room temperature before baking.
The acidity and tannin of a Chianti Classico will cut the richness of the sauce. Look for a full-bodied example.