Preheat oven to 400 degrees. Wash 4 large russet potatoes and wrap in aluminum foil. Bake until soft, about 1 hour. Actual baking time depends on size of potatoes and for the largest potatoes can take up to 2 hours.
When potatoes are done, remove from oven and set aside, still wrapped in foil. Prepare the chicken by pounding out the 2 boneless skinless chicken breasts with a meat tenderizer until 1/4 inch thick.
Prepare the dredging stations. Whisk together 1 cup buttermilk and 1 large egg and place in a shallow bowl. In another shallow bowl, combing 1 1/2 cups all-purpose flour, 1 tablespoon cornstarch, 1 tablespoon paprika, 1 tablespoon salt and 1 teaspoon black pepper.
Pour enough oil in a heavy skillet to generously cover the bottom of the pan. Heat to 350 degrees.
Working one at a time, immerse the chicken into the buttermilk mixture, then transfer it to the flour mixture to coat both sides. Place in hot oil and fry 2 minutes per side. Remove and place on a paper-towel lined tray to drain.
Unwrap the potatoes from the aluminum foil. Use a sharp knife to cut a slit in only the top of each potato. Squeeze the ends together to force the slit to bulge open. Use a spoon to scoop out the flesh of the potato into a mixing bowl, leaving enough on the edges for the skin to hold its shape.
Stir 4 tablespoons butter and 1 cup shredded cheddar cheese into the mixing bowl with the potato. Season with salt and pepper. Scoop the mixture back into each of the potato skins.
Slice the fried chicken into strips. Stuff some of the chicken into the potato.
Top stuffed potatoes with sour cream and chives, and remaining chicken.