In a 4-quart Dutch oven, cook beef, onion and celery over medium heat until meat is brown and vegetables are tender. Drain well; return to Dutch oven.
Stir in 1 can of the beef broth, undrained tomatoes, mixed vegetables, steak sauce, Worcestershire sauce, salt, and pepper. Bring to boiling; reduce heat. Cover and simmer 20 minutes.
In a medium mixing bowl, whisk together remaining can of beef broth and the flour. Stir into mixture in Dutch oven. Cook until thickened and bubbly. Cook and stir 1 minute more.