Season chicken breasts with salt and milled pepper. Heat olive oil in a large skillet until hot but not smoking. Saute chicken on medium heat for 3 to 4 minutes. Turn and saute 4 to 5 additional minutes. Remove chicken and set aside.
Add fennel and garlic to pan and cook on medium-low heat, stirring often, for 5 minutes or until the fennel strips are translucent. Add chicken broth and bring to a boil. Reduce heat and add lemon juice, sugar, lemon peel, the halved grape tomatoes (this variety is sweeter than standard cherry tomatoes) and the capers, which are slightly dill-flavored.
Simmer for 5 minutes, returning chicken to pan for the last minute to heat through. Divide chicken among 4 serving plates and top with the fennel and sauce. Good served over linguine, angel-hair pasta or rice.