In a small, heatproof bowl, soak the porcini in the boiling hot water until softened, about 15 minutes. Drain and coarsely chop the porcini.
In a large saucepan, melt 2 tablespoons of the butter in 2 tablespoons of the olive oil. Add the white mushrooms and the porcini, season with salt and white pepper and cook over moderately high heat, stirring occasionally, until browned, about 10 minutes. Transfer the mushrooms to a bowl.
Add the remaining 2 tablespoons each of butter and olive oil to the saucepan. Add the prosciutto and shallots and cook over moderately low heat, stirring occasionally, until softened, about 6 minutes. Using kitchen string, tie the bay leaf, rosemary, sage and thyme sprigs into a bundle. Add the herb bundle and the celery root to the saucepan and cook, stirring occasionally, until the celery root is crisp-tender, about 6 minutes. Add the leeks and cook until the celery root is tender, 5 minutes longer.
Return the mushrooms to the saucepan and stir over moderate heat until hot, 2 minutes. Add the Marsala and cook until evaporated, 5 minutes. Pour in 2 cups of the chicken stock and simmer over moderate heat for 10 minutes. Add 1/2 cup of the cream; simmer for 3 minutes longer, until the mixture has reduced to 5 cups. Season the ragù with salt and white pepper.
In another large saucepan, combine the remaining 2 cups each of cream and chicken stock and bring to a boil. Simmer over moderate heat until reduced to 3 cups, about 10 minutes. Off the heat, whisk in 1 1/2 cups of the Parmigiano-Reggiano cheese. Whisk in the eggs. Transfer the sauce to a blender and puree until smooth. Season with salt and white pepper. Set 1/2 cup of the sauce aside.
Bring a very large pot of salted water to a boil and fill a large bowl with ice water. Cook the lasagna noodles until just barely al dente; drain and transfer to the ice water to cool. Drain the noodles and pat dry.
Preheat the oven to 350°. Butter a 9-by-13-inch baking dish. Line the bottom with a single layer of noodles, overlapping them slightly; reserve the best noodles for the top layer. Spread one-fifth of the vegetable ragù on top, followed by one-fifth each of the sauce, mozzarella and basil. Repeat to make four more layers, ending with a layer of noodles (you won't use all the noodles). Top with the reserved 1/2 cup of sauce and sprinkle with the remaining 1/4 cup of Parmigiano-Reggiano cheese.
Bake the lasagna until bubbling and golden, about 1 hour. Let the lasagna rest for 20 minutes before cutting into squares and serving.
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