Preheat and prepare the pan: Preheat the oven to 400°F with the rack in the middle. Grease an 8×8-inch baking pan and set aside.
Mix the dry ingredients: In a large bowl, stir together the flour, cornmeal, baking powder, salt, and sugar until evenly combined.
Add the wet ingredients: Add the eggs one at a time, mixing well after each addition. Stir in the melted butter, sourdough discard, and milk. Mix just until combined.
Bake the cornbread: Pour the batter into the prepared pan and smooth the top. Bake for 20–25 minutes, or until lightly golden and a toothpick inserted in the center comes out clean.
Cool and serve: Let the cornbread cool for about 5 minutes before slicing. Serve warm with butter and honey.