Preheat oven to 425 degrees. Combine dry ingredients in a large bowl. Whisk together remaining ingredients (except for butter) and stir into the cornmeal mixture just until combined. Stir in melted butter and pour batter into a hot 10-inch cast iron skillet or baking dish.
Bake for 25 to 30 minutes or until golden and cornbread pulls away from sides of skillet. Invert cornbread onto a wire rack and cool before cutting into cubes.
Preheat oven to 350 degrees. Cut cooled cornbread into 1- to 2-inch cubes and place in baking dish lightly coated with cooking spray.
Cook butter with thyme over medium heat, stirring constantly, until butter begins to lightly brown. Remove from heat and which in milk, sugar, salt, and eggs. Pour mixture over cornbread and let stand 10 minutes. Sprinkle with the Parmesan cheese.
Cover with aluminum foil and bake for 30 minutes. Uncover and bake 30 more minutes, or until light golden brown. Let stand 10 minutes before serving.