Place a large saute pan over medium-high heat and 1/4 cup olive oil. When it begins to shimmer, add half of shallots and cook stirring occasionally, until translucent. Add mushrooms and toss to coat, cook until begin to color (12-15 minutes). Add vermouth and deglaze pan cook down for 5-7 minutes. Put in large bowl and set aside.
Meanwhile, in another bowl, toss radicchio with remaining 1/4 cup olive oil. and season with salt and pepper to taste. Spread on cookie sheet and place in oven for 15 minutes. Combine with mushrooms and set aside.
Make the béchamel: Place a sauce pan over medium heat and melt butter. When it foams, add remaining shallots and cook until translucent. Add garlic and cook stirring for 1-2 minutes. Sprinkle flour over top and cook slowly to make a dark roux, about ten minutes. Whisk in milk and continue whisking until sauce thickens. Add nutmeg and season to taste with salt and pepper.
Reserve a cup of béchamel. Pour rest over mushrooms mixture and stir to combine. Add truffle oil and stir to combine.
Assemble lasagna: spread reserved béchamel across bottom of 9 by 13 inch baking dish. Place layer of noodles in sauce, do not overlap. Spread layer of 1/3 of mushroom mixture on top of pasta and follow with shredded cheese. Put another layer of pasta above cheese, one third of mushrooms and layer of cheese. Put another layer of pasta above cheese, one third of mushrooms and layer of cheese. Cover with a buttered or oiled sheet of tin foil and place in oven for 45 minutes. Remove foil and cook until top is golden brown and bubbling, about ten minutes.