For filling, In a skillet over medium heat melt butter in olive oil. Add onions, bell peppers, zucchini, garlic, and mushrooms and saute for about 7 minutes, until softened and browned around the edges. Add basil, salt, pepper, and transfer to a bowl to cool. Filling can be made a day ahead and chilled in a covered container.
For sauce, in a 3-quart heavy saucepan melt butter over moderately low heat and whisk in flour. Cook the roux, whisking for 3 minutes and stir in milk. Bring sauce to a boil, whisking constantly, and simmer, whisking constantly for 3 minutes.
Stir in mustard, salt, and 1 1/2 cups of Asiago cheese (a little bit at a time). Remove sauce from heat and let cool slightly. Sauce may be made one day ahead and chilled in the fridge. Sauce must have a layer of waxed paper on top to prevent a skin from forming, in a closed container. Sauce should be warmed to room temp before assembling lasagna.
To make lasagna, preheat oven to 375 degrees and butter a 13"x9" baking dish.
To assemble, spread 1 1/4 cups of sauce in baking dish and cover with 3 pasta sheets, making sure they don't touch each other. Spread one third filling over pasta sheets in dish and top with 3 more pasta sheets, gently pressing down layers to remove air pockets. Top pasta sheets with 1/3 mozzarella cheese. Continue layering in the way with sauce, pasta, filling, and cheese, ending up with a layer of mozzarella on top. (Dish will be full to the brim.) Spread remaining sauce over top and sprinkle with 1/2 cup Asiago cheese.
Place 13"x9" dish on top of a foil-lined cookie sheet, to catch and spills or drips from the pan. Bake in the middle of a preheated 375 degree oven until bubbling and golden, about 45-55 minutes. Remove and let cool for 20 minutes before cutting and serving. Leftovers can be refrigerated and reheated easily. Enjoy!