Bring broth to a boil in a large saucepan. Stir in grits, and return to a boil. Cover, reduce heat, and simmer 7 minutes or until grits are thickened, stirring twice. Stir in cheese. Remove from heat.
Spoon grits into a greased 13" x 9" baking dish. Cover and chill until firm. Unmold onto a large cutting board, sliding knife or spatula under grits to loosen them from dish. Cut grits into 3/4" cubes. Place in a single layer on a large greased rimmed baking sheet or jellyroll pan.
Bake at 450° for 20 minutes; turn grits, and bake 10 to 12 more minutes or until crisp and browned.
Meanwhile, cook sausage in a large skillet, stirring until it crumbles and is no longer pink; drain.
Melt butter in a large skillet over medium heat; add celery, garlic, and onion. Sauté 5 minutes or until tender. Stir together onion mixture, sausage, and grits croutons, tossing gently. Drizzle egg over mixture; add parsley, stirring gently. Spoon dressing loosely into a greased 11" x 7" baking dish.
Bake, uncovered, at 350° for 35 to 45 minutes or until browned.