1. Cook pasta in a large pot of generously salted water until al dente. Reserve 1 1/2 cups pasta water; drain.

2. Season chicken with salt and pepper. Melt 1 tablespoon butter in a large skillet over medium-high heat. Brown chicken, flipping once, until cooked through, 6 to 8 minutes; transfer to a plate. Reduce heat to medium; melt remaining 1 tablespoon butter. Cook shallots and mushrooms, stirring, until golden brown, about 8 minutes. (If skillet browns too quickly, add a little pasta water.) Add wine, bring to a boil, and reduce by half. Slice chicken; add to pan with cream, Parmesan, 1 cup pasta water, and pasta. Toss until sauce coats pasta, adding more pasta water if needed. Add celery and parsley; toss to combine. Serve with Parmesan.