Keeping knife parallel to the counter, slice each breast into two thinner pieces. Place chicken breast between 2 sheets of plastic wrap and pound with a meat mallet (or bottom of a skillet) until ¼ inch thick. Unwrap and sprinkle with salt and pepper.
Place 1 slice of ham lengthwise on each chicken slice; divide goat cheese and 1 1/2 tablespoons pesto evenly among slices. Roll slices and tie with kitchen string.
Dredge chicken in flour. Season with salt and pepper.
Heat butter and olive oil in a large sauté pan over medium high heat. Add chicken and cook for 4-5 minutes until browned on all sides and cooked through to 165 degrees F. Remove chicken from pan and and remove strings.
Add garlic to pan and cook for 1 minute. Add wine, lemon juice and 1 tablespoon pesto and bring to a simmer. Ladle sauce over chicken and serve.