Heat oil in a large Dutch oven over medium heat. Add spinach and garlic; sauté 1 minute or until spinach begins to wilt. Remove from heat; set aside.
Place each piece of chicken between 2 sheets of heavy-duty plastic wrap and flatten to 1/4-inch thickness using a meat mallet or rolling pin. Sprinkle 1/2 teaspoon pepper and 1/4 teaspoon salt over chicken. Spoon spinach mixture evenly down center of each breast half, leaving a 1/2-inch border. Arrange bell pepper strips crosswise over spinach, and roll up each breast half jelly-roll fashion. Tuck in ends of chicken, and secure each roll with wooden picks; set aside.
Combine broth and next 7 ingredients (broth through tarragon) in pan; bring to a simmer, and cook 5 minutes. Add chicken rolls, carrots, Brussels sprouts, onions, and potatoes; simmer 20 minutes or until chicken is done and vegetables are tender. Discard peppercorns, bay leaves, and herb sprigs.
Remove wooden picks from chicken rolls, and thinly slice each roll. Arrange 1 sliced chicken roll, 5 carrots, 3 Brussels sprouts, 2 onions, and 4 potato slices in each of 4 large shallow bowls. Ladle 1/2 cup cooking liquid into each bowl, and sprinkle each with 2 tablespoons parsley.