Sprinkle meat with thyme and salt and pepper. Heat 2 tablespoons oil in heavy large deep skillet over high heat. Add 1/3 of meat and stir just until meat is no longer pink, about 1 1/2 minutes. Using slotted spoon, transfer meat to medium bowl. Repeat with remaining meat in 2 batches. Heat 1 tablespoon oil in same skillet. Add onion and saute over medium-high heat until brown, about 8 minutes. Using slotted spoon, transfer onion to bowl with meat. Add remaining 1 tablespoon oil to skillet, then mushrooms and garlic. Cover and cook until mushrooms are tender, stirring occasionally, about 4 minutes. Uncover and cook until liquid evaporates, about 3 minutes. Sprinkle mushrooms with 2 tablespoons flour; stir 1 minute. Return meat and onions with accumulated juices to skillet. Add tomatoes, broth, and cayenne; bring to boil. Reduce heat to medium-low. Simmer uncovered until meat is tender, about 5 minutes. Serve beef-mushroom mixture over egg noodles. Sprinkle with parsley.