Adjust oven rack to middle position and heat oven to 200 degrees. Spread flour in a shallow dish. Pound thicker ends of chicken breasts between 2 sheets of plastic wrap to uniform ½-inch thickness. Pat dry with paper towels, sprinkle with herbes de Provence, and season with salt and pepper. Working with 1 chicken breast at a time, dredge in flour to coat, shaking off any excess.
Heat 2 tablespoons oil in 12-inch skillet over medium-high heat until just smoking. Place chicken in skillet and cook, turning as needed, until golden brown on both sides, about 10 minutes. Transfer chicken to oven safe plate and keep warm in oven while preparing vegetables.
Heat 2 teaspoons oil in now-empty skillet over medium-high heat until shimmering. Add zucchini and squashes and cook until well browned, about 10 minutes. Stir in tomatoes and capers and cook until tomatoes are just softened, about 2 minutes. Off heat, stir in basil and 1 remaining tablespoon oil. Season with salt and pepper to taste. Serve chicken with vegetables and salsa verde.
Pulse parsley, bread, capers, anchovies, garlic, and ¼ teaspoon salt in food processor until finely chopped, about 5 pulses. Add lemon juice and pulse briefly to combine.
Transfer mixture to a medium bowl and slowly whisk in oil until incorporated. Cover and let sit at room temperature for at least 1 hour to allow flavors to meld. Season with salt and pepper to taste. (salsa verde can be refrigerated for up to 2 days. Bring to room temperature and whisk to recombine before serving.