1. Lay chicken out on a work surface, season all over with salt and pepper. Arrange 1/4 cheese, 1 tomato, and 2 basil leaves over half each scallopini. Fold open ends over filling and secure each with a toothpick. Place flour in shallow plate with a lip, and dredge chicken. Transfer to a wax paper-lined plate.
2. Heat 3 tablespoons oil in a large skillet over medium-high heat. Add chicken and cook, turning once, until lightly browned and cooked through, about 5 minutes. Divide chicken among plates and discard oil.
3. Return skillet to high heat. Add vinegar, bring to a boil while scraping up any browned bits, and cook until syrupy. Add broth, bring to a boil, and cook until mixture reduces to a glaze. Remove skillet from heat and whisk in remaining 1 teaspoon oil. Pour glaze over chicken and serve.
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