Preheat the oven to 350°F. Spray 36 madeleine-plaque molds or one 9-by-11-inch baking pan with nonstick cooking spray or grease with butter.
In a large bowl, stir together the flour, cornmeal, baking powder, baking soda, salt, and white pepper.
In a medium bowl, whisk together the eggs and sugar until thickened and smooth. Add the melted butter and whisk until incorporated. Then add the sour cream and milk and whisk until smooth.
Add the egg mixture to the flour mixture and stir until just until combined.
Add the cheese and scallions and mix gently just until evenly incorporated.
Do not overmix. Using a spoon, fill each prepared madeleine mold half full with batter. Or, pour the batter into the prepared baking pan.
Bake for 12 to 15 minutes, or until the corn bread is golden. Let cool in the pan(s) on a wire rack, then invert the madeleine plaques onto the rack. If baked in a baking pan, cut into squares and remove from the pan. Serve warm or at room temperature.