How to Make It
Cut the salmon crosswise into 1-inch strips. In a medium saucepan of boiling water, poach the salmon until just cooked through, about 2 minutes. Drain on paper towels and break into large flakes.
Roast the red bell pepper directly over a low gas flame or under the broiler, turning occasionally, until lightly charred all over, about 15 minutes. Transfer to a bowl, cover with plastic wrap and let steam for 15 minutes. Peel, core and seed the pepper. Cut into 1/4-inch dice.
In a mini food processor, puree 1 tablespoon of the dill with the lemon juice and shallot. In a slow, steady stream, add 2 tablespoons plus 2 teaspoons of the olive oil and process until emulsified. Season with salt and pepper.
In a saucepan fitted with a steamer basket, bring 1 inch of water to a boil. Add the potatoes and cook until just tender, about 4 minutes. In a medium bowl, toss the potatoes with the roasted bell pepper, flaked salmon, horseradish and the remaining 2 tablespoons of dill. Season with pepper.
Heat a large cast-iron skillet. Add the remaining 2 tablespoons of olive oil and heat until shimmering. Add the hash mixture in an even layer and fry over moderately high heat until golden and crisp on the bottom, about 3 minutes. Scrape up the hash and turn; continue frying until golden and crisp all over, about 8 minutes longer. Season with salt and pepper and transfer to plates. Top the hash with the poached eggs, drizzle with the vinaigrette and garnish with the salmon caviar. Serve at once.