Heat a heavy saucepan. Heat 1 tablespoon of the olive oil and sweat the shallots and garlic until translucent. Add 1/2 of the asparagus scraps and deglaze with vermouth. Reduce to dry and add the chicken stock. Reduce by 3/4 and place in a blender. Add remaining asparagus scraps and blend. Add olive oil in a slow, steady stream and finish with spinach. Season and strain. Keep at room temperature.
Blanch the asparagus and chill. Drain well and reserve on a cloth. Place 2 by 3 squares of blanched pasta on a surface dusted with grated Parmigiano. Place trimmed asparagus on pasta and roll tightly. About 1 1/2 inches of the tip of the asparagus should be exposed. Repeat process and group the asparagus cannelloni in 3's. Cover exposed end with tin foil and brush pasta liberally with melted butter. Add additional cheese. Keep refrigerated until needed.
Heat a saute pan with olive oil and cook chanterelles, then add garlic puree and spinach. Wilt spinach and season. Gratin cannelloni under broiler until golden brown. Remove foil and place cannelloni on bed of spinach. Drizzle sauce around and arrange mushrooms around. Sprinkle with salt and pepper. Garnish with chervil. Serve hot.