Bring 8 cups water to a boil. Remove from heat and add morels. Cover and set aside for 30 minutes. Using a slotted spoon, transfer mushrooms to a cutting board and cut each one into small pieces.
Return mushroom to heat, adding additional water sufficient to cook pasta; bring to a boil. Season water with salt, add spaghetti; cook until al dente, about 13 minutes. Drain pasta when it reaches desired level of doneness.
Meanwhile, heat oil and butter in a 12-inch skillet. Add garlic and shallots; cook, stirring occasionally, until golden, about 8 minutes. Add morels, asparagus and stock, bring to a boil. Cook, covered, until asparagus is tender, about 3 minutes. Uncover, add cream, gently cook for 3 minutes.
Remove from heat and add pasta, lemon juice and zest, salt, pepper, and cheese. Toss until evenly combined. Sprinkle with more cheese, if desired.