Heat the oven to 400°F and grease a 9-by-13-inch dish. Brown the beef, onion and carrots in a large skillet and drain.
Stir in 2 tablespoons flour for 1 minute. Add broth, Worcestershire, 1/2 teaspoon salt and pepper and simmer until thick; fold in peas and green beans and spread the hot filling in the dish.
Whisk the remaining 2 cups flour, baking powder and remaining salt. Cut in the cold butter until pebble-size, then stir in milk just until a shaggy dough forms.
Drop the biscuit dough in 12 even mounds over the hot filling, leaving small gaps for steam.
Bake for 30 to 35 minutes, until the biscuits are deep golden and cooked underneath. Rest 10 minutes.