1. Preheat oven to 350°. Lightly grease a 3-qt. baking dish with cooking spray. Cut cornbread into 2- to 3-inch pieces, and place in prepared dish. (You should have about 10 cups cornbread.) Cook butter and thyme in a 2-qt. saucepan over medium heat, stirring constantly, until butter begins to turn golden brown. Remove pan from heat, and whisk in milk and next 2 ingredients until sugar melts. Whisk in eggs; pour mixture over cornbread. Let stand 10 minutes. Sprinkle Parmesan cheese over cornbread.
2. Bake, covered, at 350° for 30 minutes. Uncover and bake 30 more minutes or until light brown and set. Let stand 10 minutes.