Rinse chicken and pat dry. Slice crosswise into 1/4-inch-thick strips.
Put spinach in a large, shallow serving or mixing bowl. In a small bowl, mix vinegar, 3 tablespoons olive oil, capers, and mustard.
Set a 10- to 12-inch frying pan over high heat. When pan is hot, add remaining tablespoon olive oil and the chicken. Stir often until chicken is no longer pink in center of thickest part (cut to test), about 4 minutes. Add onion and garlic and stir until onion begins to soften slightly, about 1 minute. Add vinaigrette and stir until hot, about 30 seconds. Pour over spinach.
Sprinkle tomatoes on top and mix; add salt and pepper to taste.