Make the hash: Put the potatoes into a small pot, cover with water, and bring to a boil over medium-high heat. Cook the potatoes until fork tender, about 10 minutes, and drain them well. Heat the olive oil in a medium skillet over medium heat. Add the onions and cook until softened, about 5 minutes. Add the salmon, cooked potatoes, and dill. Raise the heat to high and cook an additional 10 minutes, shaking the pan occasionally, until the potatoes begin to brown. Remove from heat. Season with salt and pepper.
Make the pancake: Mix the flour, salt, pepper, and sugar in a bowl. Add the beaten eggs, oil, and chives and whisk until the batter is smooth. Heat a medium nonstick skillet over medium heat. When hot, pour the batter into the skillet about 1/4-cup at a time. Flip the pancakes when small bubbles form on the top and the bottom is golden brown. Repeat with the remaining batter.
Make the egg: Using the same pancake skillet, melt a tablespoon of butter over high heat and crack the eggs into the skillet. Allow to become firm before flipping. Cook 30 more seconds
To serve: Place a pancake on a plate, top with some of the hash, and slide an egg onto the hash. Garnish with a sprig of fresh dill.