Preheat the oven to 400°F (200°C) and grease a 9-inch square baking pan.
In a large bowl, combine cornmeal, flour, baking powder, baking soda, and salt. Mix thoroughly.
In another bowl, whisk together buttermilk, egg, and melted butter until well combined.
Pour the wet mixture into the dry ingredients, stirring gently until just combined.
Fold in the shredded cheddar cheese and chopped jalapeños, mixing until just incorporated.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 20-25 minutes, or until the top is golden brown and a toothpick inserted in the center comes out clean.
Let the cornbread cool in the pan for 10 minutes before transferring to a wire rack to cool completely.
Slice and serve warm, optionally with butter or honey for added flavor.
