In a small bowl, soak the dried mushrooms in boiling water until softened, about 15 minutes. Drain and coarsely chop.
Heat a saucepan over medium-high heat, and melt 2 tablespoons of the butter with 2 tablespoons of the olive oil. Add the dried porcini and button mushrooms to the pot and sauté, stirring occasionally, about 10 minutes, until browned. Transfer to a bowl and set aside.
Add the rest of the butter and olive oil to the saucepan, then add the prosciutto and chopped shallots. Cook over medium-low heat, stirring occasionally, until softened, 5-6 minutes. Tie the bay leaf, rosemary sprigs, sage leaves and thyme sprigs together with kitchen twine. Toss the herb bundle and diced celery into the saucepan, and cook about 6 minutes, stirring occasionally. Add the leeks, and cook until celery root is tender, 5-7 minutes.
Raise the heat to medium, and return the mushrooms to the pot and cook until heated through, about 2 minutes. Pour the Marsala into the pot and simmer, stirring occasionally, until the liquid has evaporated, 5 minutes. Add 2 cups of the chicken stock, stir, and bring to a simmer. Simmer for 10 minutes. In a steady stream, pour in 1/2 cup of the heavy cream. Simmer for 3 more minutes. Season with salt and pepper and remove from heat. Remove the bundle of herbs and discard.
Heat the remaining 2 cups of chicken stock and 2 cups of cream in a separate large saucepan. Bring to a boil over medium heat and simmer 10 minutes, until the liquid has reduced to 3 cups. Remove from heat. Whisk in 1 1/2 cups of the grated Parmigiano-Reggiano. Whisk in the beaten eggs. Continue to whis for 1-2 minutes until the sauce is smooth and the cheese has melted, Season with salt and pepper. Remove 1/2 cup of the sauce, and set aside.
Bring a large pot of salted water to a boil. Fill a large bowl with ice water. Cook the lasagna until just barely al dente, then drain and transfer the noodles to the ice water. Drain.
Prreheat the oven to 350˚F.Lightly butter a 9x13-inch baking dish. Arrange a single later of slightly overlapping noodles int he bottom of the baking dish. Spread 1/5 of the celery root ragu on top of the noodles, then 1/5 of the white sauce, 1/5 of the mozzarella, and a sprinkling of basil leaves, Repeat with four more layers, ending with a layer of noodles (you'll have some noodles left over). Top with the reserved 1/2 cup of whote sauce, then with the last 1/4 cup of Parmigiano-Reggiano.
Bake until bubbling and golden, about 1 hour. Allow to rest at room temperature for 20 minutes before serving.