Cook pasta: Cook pasta according to package directions in a large Dutch oven, adding salt once water comes to a boil. Drain pasta, reserving 1 ½ cups cooking water.
Cook tomatoes and aromatics: Wipe Dutch oven clean, and add olive oil. Heat oil over medium-high. Add tomatoes to hot oil, and cook, stirring often, until slightly softened, about 2 minutes. Add shallot, and cook, stirring often, until softened, about 2 minutes. Stir in garlic; cook, stirring constantly, 1 minute.
Complete pasta: Add chicken, pasta, and herb pesto to coat. Add ½ cup of reserved cooking water, stirring to combine. Gradually stir in remaining cooking water, ¼ cup at a time, until desired consistency is reached.
Add garnishes, and serve: Transfer to a serving platter, and sprinkle with basil and mozzarella. Serve immediately.
