Place the eggs in a large pot and cover them with cool tap water. Bring the water to a boil, lower the heat, and simmer for 5 minutes. Turn off the heat and allow the eggs to sit in the water for another 5 minutes. Drain, then fill the pot with cold water. To peel the eggs, tap each end on a board, then roll the egg between your hand and the board to crackle the shell. Peel under running tap water and allow the eggs to cool to room temperature.
Place the eggs in the bowl of a food processor fitted with a steel blade. Pulse the processor 10 to 12 times to break up, but not puree, the eggs. Transfer the chopped eggs to a bowl and add the mayonnaise, mustard, dill, salt, and pepper. Combine lightly with a fork.
Toast or grill the bread. Lay 1 slice of salmon on each piece of bread, spread on the egg salad, then garnish with a sprig of dill. Serve at room temperature.
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