Fill the sink with cold water while you put 2 tablespoons of the oil in a heavy skillet large enough to hold all the chicken in one layer without crowding. Put the skillet over medium heat. Season the chicken with salt and pepper. Put the flour on a pie plate or wax paper. Press a slice of prosciutto against each of the breast halves, then press the breast halves in the flour, shaking off any excess. (If the butcher hasn’t done so, pound the chicken between two sheets of aluminum foil or wax paper with the side of a cleaver or a meat pounder before seasoning.) Raise the heat under the skillet to medium-high and add the chicken, prosciutto side down. Cook for 5 minutes on one side.
While the chicken cooks, cut the arugula crosswise into 1/2-inch wide strips, discarding the stems. Wash the arugula briefly but vigorously in the sink to remove grit. Then spin dry in a salad spinner. Remove any excess moisture with paper towels. Cut the tomato in half lengthwise, gently squeeze out some of the juice and seeds, and cut each half into thin wedges, removing the core as you do.
Turn the chicken over and cook for 3 minutes. Meanwhile, chop the basil. Put the remaining 2 tablespoons of oil, vinegar, and salt and pepper to taste in a small bowl and mix well with a whisk. Chop the sage. (If using dried sage, crush with your fingers.)
Sprinkle the sage on top of the chicken, add the wine, and cook for 1 to 2 minutes, just until the chicken is cooked through. (Cut into one piece to check if you’re not sure.) Remove the chicken to a serving platter. Raise the heat to high while scraping the bits from the bottom with a wooden spoon. Cook for 1 to 2 minutes or until reduced and lightly thickened.
While the chicken finishes cooking, put the arugula and tomato in a large mixing bowl, sprinkle with the basil and toss with the dressing. Pour the sauce over the chicken and serve with the salad.
