Preheat oven to 350°.
To prepare the chicken, place each chicken breast half between 2 sheets of heavy-duty plastic wrap; pound to 1/4-inch thickness using a meat mallet or rolling pin. Discard plastic wrap. Top each chicken breast half with 1 slice prosciutto, 1 slice cheese, and 1/4 cup arugula, leaving a 1/4-inch border around edges. Fold in half, pinching edges together to seal; sprinkle with salt and pepper. (The chicken can be prepared up to a day ahead and refrigerated at this point.)
Dredge chicken in flour, shaking off excess. Heat oil in a large nonstick skillet over medium-high heat. Add chicken; cook 5 minutes on each side. Place chicken in a shallow baking pan; bake at 350° for 5 minutes or until done. Keep warm.
To prepare sauce, add shallots to skillet; sauté 4 minutes over medium-high heat or until browned. Add tomato paste; cook 1 minute, stirring constantly. Stir in wine; bring to a boil over high heat. Cook until reduced to 1 cup (about 6 minutes). Add broth; bring to a boil. Cook until reduced by half (about 8 minutes).
Combine water and cornstarch in a small bowl, stirring with a fork until smooth. Add cornstarch mixture to sauce; bring to a boil. Cook 1 minute, stirring constantly.