Preheat the oven to 400 degrees and place a wire rack inside of a rimmed baking sheet. Set aside.
Combine ground beef, beaten egg, mustard, panko, salt, pepper, onion, beets and capers in a large bowl. Knead with your hands to combine. Shape into 4 patties and make a shallow divot in the center of each patty.
Heat 1 tablespoon of butter in a large skillet (preferably nonstick) over medium high heat. Once the foaming subsides, add the patties and cook until nicely browned, about 3 minutes per side. Place the patties on the wire rack prepped in step 1 above. Transfer the baking sheet to the oven and bake until patties are firm and cooked through, about 8-12 minutes (a thermometer inserted into each patty should read 160 degrees F)
Transfer the burgers to a plate and tent with foil. Rest for at least 5 minutes. Meanwhile, make the fried eggs: Heat the remaining tablespoon of butter in a nonstick skillet over medium. Crack 4 eggs into the skillet, salt and pepper the eggs and cover. Cook until white and yolks are just set, about 3 minutes (or until eggs are done to your liking), adjusting the heat as necessary. Top each burger with a fried egg and a sprinkle of chopped fresh dill and serve.