For the vinaigrette: place vinegar, lemon juice, shallot, dill, salt and pepper in a blender and blend until smooth. With the motor running, slowly add the oil until emulsified. Transfer the vinaigrette to a bowl or squeeze bottle.
For the hash: Bring 4 cups of water to a simmer in a medium saute pan, add the salmon and poach until just cooked through, about 8 minutes. Remove the salmon, flake with a fork and transfer the mixture to medium bowl. Add the roasted pepper, potatoes, horseradish, salt and pepper and dill and gently mix until combined.
Heat the olive oil in a large, nonstick saute pan over high heat. Add the hash mixture and cook, stirring occasionally until golden brown and crisp all over. Divide the hash among 4 plates and top each with a poached egg and a tablespoon of salmon caviar. Drizzle with the dill vinaigrette. Garnish with fresh dill.