Cook pasta according to package directions, with 1/2 teaspoon sea salt dissolved in the water.
Drain and rinse with cool water. Toss with 1 tablespoon olive oil. Set aside.
Rub chicken breasts with 1 1/2 tablespoons olive oil, salt, and pepper. Add another tablespoon of olive into a pan over medium high heat. When hot, add chicken breasts, and let cook for 5 minutes before flipping. Cook for another 5-7 minutes, until juices are running clear and no pink remains. Remove chicken and let rest on a plate.
In the same pan, add 1 1/2 tablespoons olive oil and bring pan to medium heat. Add tomatoes and stir to coat, with 1/4 teaspoon sea salt. Cook for 8-10 minutes, until tomatoes are blistered and burst. Remove and set aside.
To make walnut parmesan crumble, add walnuts to food processor and pulse 8 times. Add all other ingredients, except olive oil, and pulse 10 times. Scrape down sides. Slowly stream in olive oil as you continue to pulse. Stop when the resemble in the size of gravel.
Put pasta in a large bowl, then add 1 tablespoon olive oil, the lemon juice, arugula, and tomatoes. Toss. Then top with sliced chicken breasts and walnut parmesan crumble. Enjoy at room temperature. Will last for 3 days in refrigerator.
If you are going to eat this as leftovers, be sure to let it come to room temperature before eating otherwise noodles will be hard. You can also heat leftovers on the stove if you want something warm.