Place tofu on several layers of heavy-duty paper towels. Cover with additional paper towels; let stand 5 minutes. Place tofu in a large bowl. Add salmon to bowl; mash with a fork to crumble. Add chives and next 4 ingredients (through egg white); mix well. Divide mixture into 4 equal portions, shaping each into a 1/2-inch-thick patty. Place panko in a shallow dish; dredge patties in panko.
Heat oil in a large nonstick skillet over medium-high heat. Add patties to pan; cook 3 minutes on each side or until golden brown. Place 1 lettuce leaf on each of the bottom halves of buns; top each serving with 1 tomato slice, 1 patty, and 1 onion slice. Spread 2 tablespoons Edamame-Cilantro Pesto over each serving; top with top halves of buns.