In an extra-large skillet heat oil over medium-high heat. Add sausage; cook and stir about 3 minutes or until browned.
Add onion, celery, and garlic to skillet. Cook and stir for 2 to 3 minutes or until tender. Add sweet peppers to skillet. Cook and stir for 3 to 4 minutes or until crisp-tender.
For sauce, in a small bowl combine broth, cornstarch, brown sugar, paprika, snipped thyme, salt, and cayenne pepper. Add sauce to skillet; cook and stir until thickened and bubbly. Serve over hot cooked rice. If desired, garnish with fresh thyme sprigs.