Toss the cut-up bun with 3 tablespoons water in a large bowl; set aside 1 minute. Add the chicken, 3/4 teaspoon salt, and pepper to taste; mix with your hands until combined. Form into four 3/4-inch-thick patties. Heat the olive oil in a large nonstick skillet over medium heat. Add the almonds and cook, stirring, until lightly toasted, about 2 minutes. Set aside 1 tablespoon almonds for the salad; transfer the rest to a blender along with the oil from the pan. Do not wipe out the skillet. Increase the heat under the skillet to medium high. Add the chicken patties and cook until no longer pink, 7 to 8 minutes per side. Meanwhile, add the whole tomato to the blender and process until smooth. Add the basil and 1/4 teaspoon salt; pulse until almost smooth. Add the parmesan; pulse until combined. Season with salt and pepper. Serve the burgers on the buns; top each with a spoonful of pesto and a tomato slice. Toss the remaining pesto with the arugula, the reserved 1 tablespoon almonds and the lemon juice; serve with the burgers.