Heat the oil in a 4 1/2 qt Dutch oven or large soup pot over medium-low heat. Cut the chicken into bite sized chunks, adding them to the pot as you cut. Raise the heat to medium, add the chopped onions and cook for 2 minutes, stirring from time to time.
Add the broth and raise the heat to high. Cover, and bring to a boil. Drain the tomatoes and add them to the pot, along with the Worcestershire sauce and garlic, keeping the lid closed as much as possible. When the soup boils, reduce the heat to medium-low. Continue to cook at a slow boil until the chicken in no longer pink but tender, about 6 minutes. Remove from heat.
Stir in the half and half, pesto and salt and pepper. Let the soup rest off the heat for about 2 minutes, then serve.