Preheat the oven to 375°F. Spray an 8-inch square baking pan with non-stick cooking spray.Brown the onions. Heat the olive oil in a skillet over medium-low heat. Add the sweet onions and cook for 7 to 10 minutes, until they are light golden brown, stirring frequently. Set aside to cool.Make the cornbread batter. In a large bowl, combine the flour, cornmeal, baking powder, sugar, salt, and pepper.In a separate bowl, whisk the milk with the egg white. Add the cooled cooked onions and blend well. Add the wet mixture to the dry ingredients, stirring just until blended. Bake. Spread the batter in the prepared pan and bake for 20-25 minutes, or until the sides begin to pull away from the pan.Cool. Let the cornbread cool on a rack before loosening the sides and turning it out of the pan.