In a medium mixing bowl, mix together the flour, cornmeal, cornstarch, baking powder, baking soda, salt and pepper and set aside.
In another bowl, whisk together the buttermilk and eggs. Pour the wet ingredients into the dry ingredients and gently mix. Add the melted butter with olive oil and sea salt, cheddar, chives and shallots to the batter and stir to combine.
Preheat oven to 200°F. Preheat waffle iron to medium. Pour in as much batter as your waffle iron's instructions recommend and cook until golden. Gently remove each waffle and transfer to a baking sheet. Repeat this process until you’ve used all the batter.
Pop the waffles into the oven for 5-10 minutes to warm before serving with a drizzle of the butter with olive oil and sea salt and a fried egg on top.