Melt butter in a sauce pan over medium heat. Add flour, salt and nutmeg*one minute. Gradually stir in milk. Cook until sauce thickens and bubbles, stirring constantly. Continue cooking one more minute. Set aside.
Preheat oven to 375
Trim about 1 inch From bottoms of asparagus spears and discard steam asparagus for six minutes or until tender. Cool slightly: cut on additional inch off bottoms of Spears. Place cut pieces and oil in blender or food processor: purée until creamy. Stir asparagus purée into white sauce.
Spread 1 cup asparagus sauce in 13 x 9" baking dish; top with four lasagna sheets and another cup of asparagus sauce. Top with half of the asparagus spears, one third of both cheeses, 1 cup mushroom and garlic sauce. Top with four lasagna sheets, one cup asparagus sauce, the remaining asparagus spears and one third of both cheeses. Top with four more lasagna sheets, remaining mushroom and garlic sauce and cheese. Cover with foil; bake 45 minutes. Remove from oven; let stand 15 minutes before cutting. Serves 7-9.