In a small bowl, add all the dressing ingredients and whisk until well combined. Set aside.
Meanwhile, cook the pasta in a pot of boiling salted water, according to the package directions. Drain, rinse with cold water and transfer to a large mixing bowl. Set aside.
While the pasta is cooking, prepare the chicken. Slice the chicken breasts crosswise to create long (0.5-inch thick) strips.
In a mixing bowl, combine 1 tablespoon olive oil with the thyme, oregano, garlic, and a pinch of salt and pepper.
Add in the chicken pieces and stir to coat. Heat a large non-stick pan over medium heat. Add the chicken in a single layer, and cook for 6-8 minutes, or until cooked through and golden on the sides.
Transfer the chicken along with any accumulated juices to the pasta bowl. Add the pepper and onion, cherry tomatoes, olives, parmesan, and arugula.
Pour over the dressing and toss to combine. Season to taste with more salt and pepper, if necessary, and serve warm or chilled.