1Place chicken in a medium saucepan and add water to cover by 1 inch. Bring to a boil. Cover, reduce heat to low and simmer gently until no longer pink in the middle, 10 to 15 minutes. Transfer to a clean cutting board; shred into bite-size pieces when cool enough to handle.
2Combine pesto, mayonnaise and onion in a medium bowl. Add the chicken and toss to coat. Whisk oil, vinegar, salt and pepper in a large bowl. Add greens and tomatoes and toss to coat. Divide the green salad among 4 plates and top with the chicken salad.