Bring a large pot of water to a boil over high heat. Add the pasta and cook according to package directions. Drain pasta in a strainer.
In a medium skillet, over medium-high heat, add 1 tablespoon oil and chicken. Cook thoroughly. Season with salt and pepper.
In a food processor, add avocado, arugula, jalapeno, cilantro, garlic, lime juice, salt and pepper. Puree until smooth. Add the Parmesan cheese and puree again until it is mixed in.
Pour the avocado pesto and chicken over the pasta, then toss gently to combine. If the pasta seems too thick and sticky, add a bit of the re-served pasta water.
Add the cherry tomatoes and toss to combine. Serve warm or at room temperature, topped with additional Parmesan as desired.