Butterfly each chicken breast: place one breast flat, slice horizontally with a sharp knife to create two even cutlets; repeat with the second breast.
Make the marinade by combining 1/4 cup olive oil, 5 smashed garlic cloves, 1 tablespoon mustard, 1 teaspoon salt, and 1/2 teaspoon freshly ground black pepper in a zip-top bag or large nonmetallic bowl.
Add the chicken cutlets to the marinade, coat well, and let sit for at least 10 minutes (or up to 24 hours in the refrigerator).
Heat a large non-stick or cast-iron skillet over medium-high heat. Remove chicken from the marinade and pat dry with paper towels.
Add 1 teaspoon olive oil to the hot pan. Cook the chicken cutlets in a single layer for about 3 minutes per side, until golden and 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 (5 cloves), and 1/3 cup chicken broth.
Cook the vegetables over medium-low heat until the broth has mostly evaporated and the vegetables are tender-crisp; add a little more broth (up to 1/3 cup) if you prefer softer vegetables.
Return the chicken and any accumulated juices to the skillet, then add 1/3 cup butter. Allow the butter to melt and spoon it over the chicken and vegetables to glaze.
Serve the chicken and vegetables garnished with fresh basil.