Place all listed ingredients except olive oil in a food processor. Season with S&P as desired. Pulse to combine, stopping to scrape down the sides of the food processor as needed. With the processor running, slowly stream in the olive oil until your desired consistency is reached. Set aside, or transfer to an airtight container & store in the refrigerator for up to 1 week.
Bring a large pot of water to a boil. Once boiling, generously salt the water. Add in the pasta and cook to al dente, according to the package directions. Drain the pasta, rinsing with cold water, & set aside.
In a medium bowl, whisk together the olive oil, lemon zest & juice, & garlic. Season with S&P as desired. Toss the bell pepper, zucchini, & cherry tomatoes through the marinade. Transfer to a large plate or small baking sheet, stringing the marinated cherry tomatoes on a skewer, & set aside. Add the chicken to the bowl, tossing to coat in the remaining marinade.
Place the chicken & veggies on the prepared grill, ensuring the tomato skewers & zucchini are positioned perpendicular to the grill grate to help prevent them from falling through the grates as they cook. Grill vegetables until as softened & charred as desired - tomatoes will grill in about 2 minutes per side, while bell pepper & zucchini need 3-4 minutes per side. Grill the chicken for 4-5 minutes per side, or until fully cooked through. Remove from the grill and set aside to cool slightly.
Once cool enough to handle, chop the grilled chicken, bell pepper, & zucchini into bite-sized pieces. Add the cooked pasta, prepared lemon basil pesto, grilled chicken, grilled veggies, & mozzarella pearls to a large bowl. Toss to combine. Taste and season with additional salt or ground black pepper as needed.