In a large skillet over medium-high heat, heat oil. Season chicken with salt and pepper and cook until golden and cooked through, about 8 minutes per side. Transfer to a plate.
Add onion and peppers to skillet and cook until soft, 5 minutes. Add zucchini and cook until slightly charred, 3 minutes more, then add garlic and cook until fragrant, 1 minute more. Season with oregano.
Add broth and crushed tomatoes and simmer 10 minutes. Return chicken to skillet and top with sauce and vegetables.
Garnish with basil before serving.