Preheat the oven to 400 degrees.

Arrange the beets in a small baking dish and fill with enough water to submerge the beets 1 inch. Season with salt, cover the dish with foil, and bake in the oven for 30-40 minutes, until the beets are fork tender. Allow beets to cool enough to handle. Remove skins with a paring knife—they should come off easily. Dice beets and set aside in a medium mixing bowl.

In the meantime, while the beets are roasting, coat a large skillet with olive oil and sauté the potatoes over medium-high heat. When the potatoes are tender, but not completely cooked, add the onions and season generously with salt and pepper. Continue to sauté until the onions and potatoes are slightly caramelized, another 5-7 minutes. Add the potato-onion mixture to the beets and allow both to cool.In a small cup or bowl, scramble 1 of the eggs. Add to the beet-potato mixture along with the dill and ½ teaspoon of salt. Toss until the vegetables are fully coated.

With a paper towel, remove any scraps from the skillet so it is more or less clean, and add another generous coating of olive oil. Get the oil nice and hot, then add the hash in one layer, pressing down with your spatula to form a large cake. Cook for a few minutes until the bottom is crispy and the hash has firmed up. Flip in sections and cook until the other side is crispy. Remove the hash and split between two serving plates.

(Optional) Clean the pan, add some more oil and fry two eggs. Top each plate of hash with a fried egg and serve immediately.