For the chicken soak mixture: Stir together the buttermilk, Dijon, salt, paprika and cayenne 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, salt, mustard powder and baking soda in a bowl. Remove the chicken from the soak mixture and add to the flour mixture, turning to coat. Remove, then dip again in the soak mixture and then again in the flour mixture.
Heat 1 1/2 inches 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 in 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. Whisk together the flour, sugar, baking powder, salt and baking soda in a large bowl.
Whisk together the buttermilk, Cheddar, butter, egg yolks and scallions in a separate bowl. Add the wet ingredients to the dry and mix together. Beat the egg whites in a separate bowl until stiff. Gently fold the egg whites into the batter. Ladle the mixture into the waffle iron and cook until the waffles are golden and cooked through. Serve the chicken on top of the waffles.