In a small bowl, combine the yogurt and mint; season with salt. In a medium bowl, add the cucumber and sprinkle with salt; let stand for 15 minutes, then transfer to paper towels to drain.
Meanwhile, slice the salmon lengthwise into 1/4-inch-wide strips, cut in half crosswise, then cut into 1/4-inch pieces. In a medium bowl, beat the raw egg with 1/2 teaspoon salt. Stir in the salmon.
In a large, nonstick skillet, heat the oil over medium-high heat. Add the salmon mixture in 4 mounds; gently press to form 4 patties. Cook, turning once, for 6 minutes.
Place a salmon patty on each roll bottom; top with 2 tablespoons yogurt-mint sauce, some cucumber, sliced hard-boiled egg, spinach and a roll top.