Keep the waffles warm in a 200-degree oven while making the chicken.
Whisk together eggs, 2 cups buttermilk, and butter. In another bowl, whisk together flour, baking powder, baking soda, and 3/4 teaspoon salt. Transfer 1 cup flour mixture to a shallow bowl; fold rest into wet ingredients. Preheat a Belgian-waffle iron; brush with oil. Pour in 1 cup batter; spread, leaving a 1/2-inch border. Cook until golden brown, 8 to 10 minutes. Repeat with remaining batter.
Heat oil in a large skillet over medium-high. Season chicken with salt and pepper; dredge in reserved flour mixture. Shake off excess. Dip in remaining 1 cup buttermilk, then again in flour; repeat. Cook in batches until golden brown, 1 to 2 minutes a side. Transfer to a plate and serve, with waffles, butter, syrup, and hot sauce.