Bring a pot of salted water to a boil. Add the pasta and cook as the label directs; about 3 minutes before the pasta is done, add the green beans. Drain the pasta and beans and rinse under cold water. Toss with half of the cheese, 1 tablespoon olive oil, the lemon zest and salt.
Preheat the broiler. Puree the arugula, parsley, nuts, lemon juice, 3 tablespoons water, the remaining cheese and 1 tablespoon olive oil, and salt to taste in a food processor. Transfer to a bowl.
Place the chicken breasts between two pieces of plastic wrap and pound to about 1/2 inch thick. Transfer to a foil-lined baking sheet and season with salt. Rub all over with 1/4 cup of the arugula pesto and broil, turning once, until cooked through, 4 to 5 minutes per side.
Arrange the tomatoes cut-side up on another baking sheet and broil 2 minutes. Spread each with some pesto and sprinkle with the breadcrumbs. Broil until golden, 1 more minute. Serve the chicken, tomatoes and pasta salad with the remaining pesto.