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, or until waffle maker signals ready. Repeat with remaining batter. Keep waffles warm in a 200° oven while making the chicken.
Heat oil in a large skillet over medium-high heat. 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 and juices run clear. 4-5 minutes per side. Cooking time with vary according to thickness of chicken, temperature of oil etc.
Serve waffles and fried chicken with pure maple syrup.