Preheat a cast-iron grill pan. Brush the asparagus with olive oil and season lightly with salt and pepper. Grill the asparagus over high heat, turning, until it is lightly charred and very tender, about 6 minutes. Cut the asparagus into 1-inch lengths.
In a very large, deep skillet, heat the 3 tablespoons of olive oil. Add the shallots and cook over moderate heat until fragrant, about 1 minute. Add the shiitake, season with salt and pepper and cook, stirring once or twice, until the mushrooms are golden and tender, about 8 minutes. Add the cream and bring to a boil. Simmer until slightly reduced, about 4 minutes.
Meanwhile, in a large pot of boiling salted water, cook the pasta until al dente. Drain the pasta, reserving 3/4 cup of the cooking water.
Add the pasta to the skillet along with the asparagus, peas and grated cheese and toss well. Add the reserved pasta water and simmer, tossing, until the pasta is nicely coated. Season the pasta with salt and pepper and stir in the parsley. Serve right away.