Brown sausage in a 10-inch deep-dish ovenproof skillet over medium-high heat, stirring occasionally, 4 to 6 minutes. Remove sausage, and drain on paper towels, reserving 2 tablespoons drippings in skillet. Add onion, carrot, and garlic; cook, stirring constantly, 2 to 3 minutes.
Stir in sausage, beans, and chicken broth. Reduce heat, and simmer mixture 6 minutes. Stir in 1 tablespoon chopped fresh thyme, and remove from heat.
Stir together baking mix, milk, egg, and remaining 1 tablespoon thyme with a fork until moistened.
Pour dough evenly over top of sausage mixture in skillet.
Bake at 400° for 40 minutes or until golden brown.
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