Combine the onion, lemon juice, sugar, 1/4 cup water, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl; set aside.
Put the dill, walnuts and garlic in a food processor and pulse until coarsely chopped. With the motor running, slowly add the olive oil until the mixture comes together, scraping down the sides of the processor as needed. Transfer the dill pesto to a large bowl and stir in the lemon zest and 1/2 teaspoon each salt and pepper. Add the chicken and toss to coat.
Drain the onion and squeeze dry. Drizzle the tomatoes with olive oil and season with salt and pepper. Divide the lettuce and tomatoes among 4 pieces of bread; top each with some chicken salad, onion and another piece of bread. Serve with chips.