To prepare grits, bring water and salt to a boil in a large saucepan; gradually stir in grits. Reduce heat; simmer 30 minutes or until thick, stirring constantly. Remove from heat; stir in 1/4 cup cheese and white pepper. Stir in egg.
Spoon 1/2 cup grits mixture into each of 8 (4-ounce) ramekins or custard cups coated with cooking spray. Place ramekins in a 13 x 9-inch baking pan; add hot water to pan to a depth of 1 inch. Bake at 325° for 20 minutes. Remove ramekins from pan; cool completely on a wire rack.
To prepare sauce, bring shallots, wine, 1/4 cup broth, vinegar, bay leaf, and chile to a boil in a small saucepan over medium heat; cook until reduced to 1 tablespoon liquid (about 6 minutes). Strain mixture through a sieve over a bowl; discard solids. Return liquid to pan.
Combine 1/4 cup broth and cornstarch. Add cornstarch mixture to pan; bring to a boil. Cook 1 minute, stirring constantly. Reduce heat to low; add butter and cream, stirring with a whisk until well blended. Stir in 2 tablespoons cheese, juice, and black pepper.
Cook the ham in a large nonstick skillet coated with cooking spray over medium heat 1 minute. Add mushrooms, 1 tablespoon shallots, 1 teaspoon water, and 3 thyme sprigs; sauté 3 minutes or just until mushrooms are tender.
Loosen edges of grits with a knife or rubber spatula. Place a plate, upside down, on top of each ramekin; invert onto plates. Spoon 1 tablespoon sauce onto each plate; sprinkle ham mixture evenly among plates. Garnish with thyme sprigs, if desired.