Bring grits, salt, and 6 cups water to a boil in a large heavy saucepan over medium heat, stirring constantly. Reduce heat to low; simmer, stirring frequently, 20 to 25 minutes or until very thick. Remove from heat; stir in Parmesan cheese until melted.
Spoon grits into a 13- x 9-inch pan lined with heavy-duty plastic wrap; spread in an even layer. Cool 15 minutes. Place a dry paper towel over grits, and cover with plastic wrap. Chill 2 hours or until very firm.
Preheat oven to 425°. Turn chilled grits out onto a cutting board; remove plastic wrap and paper towel, and cut grits into 12 squares.
Whisk together egg and 2 Tbsp. water in a bowl. Combine panko and, if desired, ground red pepper in a shallow dish. Dip grits patties into egg wash; dredge in panko mixture. Place grits patties on a baking sheet coated with cooking spray.
Bake at 425° for 25 minutes or until lightly browned.
Glorious Grits: Fresh, Flavorful Recipes for Grits, Cornmeal & Polenta