Preheat the oven to 375°F (190°C). Grease an 8x8-inch baking pan or use a cast-iron skillet for extra crispy edges.
Prepare the dry ingredients: In a large bowl, whisk together the cornmeal, flour, baking powder, salt, and sugar. Set aside.
Mix the wet ingredients: In a separate bowl, beat the eggs, then add the milk and melted butter. Stir until smooth and well combined.
Add the cheese: Mix the shredded cheese (and jalapeños, if using) into the wet ingredients.
Combine wet and dry ingredients: Pour the wet ingredients into the dry ingredients, and gently stir until just combined. Be careful not to overmix, as that can affect the texture.
Bake the cornbread: Pour the batter into the prepared pan and spread it evenly. Optionally, sprinkle a little extra cheese on top for a cheesy crust. Bake for 25-30 minutes or until golden brown, and a toothpick inserted into the center comes out clean.
Cool and serve: Let the cornbread cool for a few minutes before slicing into squares or wedges. Serve warm with butter or as a side dish to your favorite meal.