Heat the oven to 400°F and grease a 9-by-13-inch dish.
Heat the oil in a large skillet. Brown the mushrooms for 6 minutes, then add the onion and beef and cook until the beef is browned. Drain excess fat.
Stir in the flour for 1 minute. Add the broth, milk, Worcestershire, salt and pepper and simmer until thickened, then fold in the green beans.
Spread the filling in the dish, sprinkle with cheddar and arrange the frozen tots in a single layer.
Bake for 35 to 40 minutes, until the tots are deep golden and the filling bubbles. Rest for 10 minutes.