Preheat oven to 400°. On a rimmed baking pan, mix potatoes with onion, olive oil, salt, paprika, and a few grinds of pepper. Bake, stirring a couple of times, until potatoes are tender, 20 to 25 minutes.
Crumble the fillet into bite-size pieces as you add it to the pan. Add dill and lemon juice and mix well, scraping up any browned bits from bottom of pan. Season hash with additional salt and pepper if you like and cover with foil to keep warm.
Bring a large, deep frying pan of lightly salted water to a boil. Reduce heat to maintain a steady simmer. Crack one egg into a glass measuring cup, hold spout close to water's surface, and let egg slip gently into water. Repeat with rest of eggs. Cook 3 minutes, or until eggs are softly set.
Divide hash between 2 plates. Using a slotted spoon, transfer 2 poached eggs onto each plateful of hash and sprinkle with pepper to taste.
Buying Eggs. Quick Facts:
Pastured Eggs. What they are: Laid by chickens that spend most of their time outside, foraging on grass and insects. Price: Up to $7 per dozen. Where to find them: Farmers' markets; farm stands; a few markets (see localharvest.org or eatwild.com for sources near you).
Commercial Eggs. What they are: Laid by chickens typically fed a blend of grains, soy beans, and vitamins; birds are often confined to cages. Price: About $3 per dozen. Where to find them: Any grocery store.
Note: Nutritional analysis is per serving and using commercial eggs.