In a medium bowl, whisk together the tartar sauce and ketchup. Reserve 3 tbsp. of the sauce in a separate cup. Add the coleslaw mix to the bowl and toss well; season with salt and pepper.
In a shallow, medium bowl, beat the eggs. Pour the breadcrumbs onto a separate plate and stir in 1/2 tsp. each salt and pepper. Working with 1 fillet at a time, lightly wet the fish under running water. Coat in the breadcrumbs, then in the beaten eggs, then once more in the breadcrumbs. Transfer to a large plate.
Preheat the broiler. In a large bowl, crumble the beef and gently mix in 1/2 tsp. each salt and pepper. Form the meat into four 4-inch patties; set aside. Toast the buns cut sides up under the broiler until golden-brown.
In a large skillet, melt 1 tbsp. butter over medium heat. Add the burgers and cook until brown on 1 side, 3 minutes. Flip, top each with 1 slice cheese, cover and cook for 2 minutes.
In a large nonstick skillet, melt the remaining 2 tbsp. butter over medium heat. Add the fish fillets and cook until lightly browned on 1 side, about 2 minutes. Flip, top each with 1 slice cheese, cover and cook for 2 minutes.
Spread the reserved 3 tbsp. sauce on the bun bottoms. Top each with a tomato slice and a burger. Top each burger with a fish fillet, some of the slaw and a bun top. Serve with the remaining slaw.