In a medium mixing bowl, stir together the flour, corn meal, cornstarch, baking powder, baking soda, salt, and pepper.
In a small bowl or glass measuring cup, whisk together the buttermilk and eggs.
Pour into bowl with dry ingredients and mix gently but thoroughly.
Pour in the melted butter and stir to incorporate.
Then stir in the cheddar and chives.
Preheat oven to 200°F with a rack in the center.
Preheat waffle iron to medium or medium-dark setting. (Waffle irons vary tremendously, so you may have to experiment a little.)
Pour in as much batter as your waffle iron's instructions recommend and cook until golden.
Gently remove each waffle from iron and place directly on oven rack. The waffles will still be slightly soft when they come out of the iron but will crisp up on the outside and stay tender on the inside while they rest in the oven.
When all waffles are cooked and have had a couple of minutes in the oven, serve with fried eggs on top and a dollop of sour cream if you like.