Place each chicken breast on a cutting board and butterfly: slice horizontally through the breast to create two even cutlets; repeat with the second breast.
Combine 1/4 cup olive oil, 5 smashed garlic cloves, 1 tablespoon mustard, 1 teaspoon salt, and 1/2 teaspoon black pepper in a zip-top bag or nonmetallic bowl.
Add the chicken cutlets to the marinade, seal, and marinate for at least 10 minutes or up to 24 hours in the refrigerator.
Heat a large non-stick pan or cast-iron skillet over medium-high heat and add 1 teaspoon olive oil once hot.
Remove chicken from the marinade and pat dry. Cook the cutlets in a single layer for about 3 minutes per side, until golden and just cooked through; transfer to a plate and tent with foil.
In the same skillet, add the broccoli florets, frozen peas, trimmed green beans, chopped garlic, and 1/3 cup chicken broth or stock.
Cook the vegetables over medium-low heat until the broth has evaporated and the vegetables are tender-crisp (add an extra 1/4–1/3 cup broth if you prefer softer vegetables).
Return the chicken and any accumulated juices to the pan, add 1/3 cup butter, and allow it to melt, spooning the butter over the chicken and vegetables to coat.
Serve the chicken and vegetables garnished with basil.