Combine broth, water, and grits in a large, heavy saucepan; bring to a boil, stirring constantly. Reduce heat, and simmer, uncovered, 20 to 25 minutes or until very thick, stirring often. Remove from heat, and add Parmesan cheese, stirring to melt cheese. Spoon grits mixture into a 13 x 9 baking pan lined with heavy-duty plastic wrap or coated with cooking spray. Cool completely. Cover and chill grits 2 hours or until very firm.
Preheat oven to 450°. Invert grits onto a large cutting board, and remove plastic wrap. Cut grits into 3/4-inch cubes. Place in a single layer on a large baking sheet or jelly-roll pan coated with cooking spray. Bake at 450° for 20 minutes; turn grits cubes, and bake 12 more minutes or until crisp and browned. Reduce oven temperature to 350°.
Cook sausage in a large skillet, stirring to crumble, until sausage is no longer pink; remove sausage from skillet, reserving drippings in skillet. Add butter to drippings in skillet, and place over medium-high heat until butter melts. Add celery, garlic, and onion; sauté 5 minutes or until vegetables are tender. Combine onion mixture, sausage, grits cubes, and parsley, tossing gently. Drizzle egg over grits mixture, and toss gently; spoon into an 11 x 7 baking dish coated with cooking spray. Bake, uncovered, at 350° for 40 to 45 minutes or until browned.