Heat oven to 325 degrees. Soak mushrooms in a small bowl of warm water until soft, 15 minutes. Drain, rinse and chop. Season meat generously with salt and pepper.
Heat olive oil in a Dutch oven over medium heat. Brown meat, about 2 minutes per side. Lift out; set aside. Add carrot, onion, celery and garlic to the pot. Cook, stirring, until softened, about 10 minutes. Add broth, wine, tomato paste, reserved mushrooms, bay leaves and thyme. Heat to a boil. Return meat to the pot. Liquid should come about two thirds the way up the sides of the roast. Cover.
Place in oven; cook, turning roast once and basting occasionally, until very tender when prodded with a fork, 2-2 1/2 hours.
Set pot on stovetop. Remove meat from pot. Shred. Discard any fat. Discard bay leaves and thyme sprig. Stir shredded meat into sauce; keep warm over low heat.
Heat a large pot of salted water to a boil. Cook noodles until tender but firm. Drain. Add noodles to meat and sauce. Toss gently.