Heat 4 tablespoons olive oil in heavy medium skillet over medium heat. Add shallots and sauté until translucent, about 4 minutes. Increase heat to medium-high. Add white wine, lemon juice and drained capers and cook until almost all liquid evaporates, about 12 minutes. Transfer shallot mixture to large bowl. Refrigerate until shallot mixture is well chilled, about 1 hour.
Using on/off turns, coarsely grind salmon fillets in processor. Add ground salmon to shallot mixture. Mix in fresh breadcrumbs, beaten eggs, dill, 1 1/2 teaspoons salt and 3/4 teaspoon pepper. Form salmon mixture into 10 patties, dividing equally. (Salmon patties can be prepared 6 hours ahead. Transfer to baking sheet, then cover with plastic wrap and refrigerate.)
Heat 1 tablespoon olive oil in heavy large skillet over medium-high heat. Working in batches, add salmon patties to skillet and cook until patties are golden brown and cooked through, about 2 minutes per side, adding more oil to skillet as needed. Serve salmon burgers on toasted hamburger buns with mayonnaise, lettuce leaves and tomato slices.