Preheat oven to 350°. Place oil in a large nonstick skillet over medium heat; add onion, celery, and carrot. Cook 10 minutes, stirring frequently. Stir in sage, thyme, salt, and pepper. Remove from heat.
Whisk butter, eggs, and broth in a large bowl. Add onion mixture, whole-wheat cubes, and cornbread crumbs; toss. Spoon into a 13 x 9-inch baking dish coated with cooking spray. Cover and bake at 350° for 20 minutes. Uncover; bake 20 minutes more, or until top is toasted.