Preheat oven to 400°F with a 12 inch cast iron skillet in the oven.
Using a stand mixer, cream 1/3 cup butter and 1/4 cup granulated sugar together on medium high speed until light and fluffy, about 2-3 minutes. Scrape sides.
In a medium sized bowl, combine 1 cup medium grind corn meal, 2/3 cup all-purpose flour, 2 teaspoons baking powder, and 1/2 teaspoon salt. In a separate bowl, combine 2 large eggs and 2/3 cup whole milk. Alternate adding small amounts of the dry ingredients and the wet ingredients to the butter mixture, each time mixing just until combined.
Pull hot cast iron skillet out of the oven and melt remaining 2 tablespoons butter in the skillet, spreading around the bottom and up the sides. Pour batter into skillet and cook for 20 minutes in 400°F oven.
Remove from oven and allow to cool slightly before removing from skillet to fully cool.
Dressing:
Cut cornbread into 1 inch cubes.
Melt 1/2 cup unsalted butter in a large skillet over medium high heat. Add 2 cups yellow onion, 2 cups celery, and 4 tablespoons fresh minced herbs (not the parsley). Stir to coat in the melted butter and then sauté until they start to turn golden brown, about 15 minutes.
Add 3 cups chicken stock and 1 teaspoon kosher salt to skillet and bring to a simmer. Remove from heat and set aside. Allow to cool enough that mixture is cool to the touch. You must allow this mixture to cool because you'll be mixing it with egg and do not want it to scramble.
Meanwhile, beat 1 large egg with 1/4 cup parsley minced in a large bowl; add cubed cornbread and gently toss to coat. Pour the cooled vegetable and broth mixture over the bread and toss. Transfer to a buttered baking dish and dot with remaining 2 tablespoons unsalted butter.
Cover and bake 30 minutes at 375°F; uncover and bake until golden, 30 more minutes.
Notes
If you like crispy dressing, there are a few things you can do:
You can heat some olive oil in a saute pan and sear the cubes of cornbread prior to mixing with remaining stuffing ingredients.
You can bake the dressing in a larger baking dish so that there is more surface area to make it crispy. The deeper and smaller the dish, the mushier the dressing will be.