Preheat your oven to 400°F (200°C). Lightly grease a 9x13 inch baking dish.
Make the filling: Melt 1/4 cup of butter in a large skillet over medium heat. Add the onion and cook until soft, about 5 minutes.
Whisk in the flour to create a roux. Cook for 1 minute, stirring constantly.
Gradually whisk in the chicken broth and milk until the mixture thickens. Stir in the cooked chicken, frozen vegetables, thyme, salt, and pepper. Heat through, then pour the filling into the prepared baking dish.
Make the cornbread crust: In a mixing bowl, combine the cornmeal, flour, sugar, baking powder, and salt. If using, stir in the shredded cheddar cheese.
In a separate bowl, whisk together the milk, egg, and melted butter. Pour the wet ingredients into the dry ingredients and mix just until combined. Do not overmix.
Spoon the cornbread batter evenly over the hot chicken filling in the baking dish. Spread it gently to cover the surface.
Bake for 35 to 40 minutes, or until the cornbread topping is golden brown and cooked through. Let it rest for 5 minutes before serving.