Preheat your oven to 375°F (190°C) and grease an 8-inch square baking pan.
Melt the butter and pour it into a mixing bowl. Blend in both the granulated and brown sugars until thoroughly combined. Add the eggs, mixing until well incorporated.
In a separate bowl, mix the buttermilk with baking soda, then stir this mixture into the butter and sugar blend.
Gradually add the cornmeal, flour, and salt to the wet ingredients, stirring until just combined. Ensure the batter remains slightly lumpy to avoid over-mixing.
Pour the batter into the prepared pan, spreading it evenly. Bake for 22-30 minutes, or until a toothpick inserted in the center comes out clean.