Special equipment: a waffle iron
For the chicken soak mixture: Stir together the buttermilk, Dijon, paprika, cayenne and salt in a large bowl. Add the chicken pieces and submerge them to coat. Allow to sit for at least 30 minutes but ideally as long as possible (up to 2 hours at room temperature or up to 8 hours, refrigerated).
For the flour mixture: Preheat the oven to 375 degrees F; set a rack in a rimmed baking sheet.
Combine the flour, baking soda, salt and mustard powder in a bowl. Remove the chicken from the soak mixture and add to the flour mixture, turning to coat. Remove, dip again in the soak mixture and then again in the flour mixture.
Heat 1 1/2 inches of oil in a very large cast-iron skillet over medium-high heat. When hot, shallow-fry the chicken, turning once, until golden brown and an instant-read thermometer inserted into the thickest part (avoid bone) registers 165 degrees F, around 15 minutes per side. Transfer the chicken to the prepared baking sheet and place in the oven to finish cooking through, if necessary.
For the waffles: Preheat a waffle iron. In a large mixing bowl, whisk together the flour, sugar, baking powder, baking soda and salt.
In a separate bowl, whisk together the buttermilk, Cheddar, butter, egg yolks and scallions. Add the wet ingredients to the dry and mix together. In a separate bowl, beat the egg whites until stiff. Gently fold the egg whites into the batter. Ladle mixture into the waffle iron and cook until the waffles are golden and cooked through.
Serve the chicken with the waffles on the side.