Using a meat mallet, pound chicken breasts to an even ¼-inch thickness.
In a large mixing bowl combine half of the wine with 1 teaspoon cornstarch, ½ teaspoon salt, and ¼ teaspoon fresh ground black pepper; whisk to combine.
Add chicken breasts to the wine mixture and toss to evenly coat. Set aside for 20 minutes.
In a separate smaller mixing bowl whisk together the chicken broth, lemon juice, remaining wine, and 1 teaspoon cornstarch. Set aside for later.
Heat 1-½ tablespoons oil in a large 12-inch skillet set over medium-high heat
Add chicken breasts to the skillet and cook for 5 minutes; flip, add 1 tablespoon butter, and cook for 4 more minutes, or until just cooked through. The thicker the chicken breasts, the longer they'll have to cook. Use an instant read food thermometer to check for doneness. Chicken breasts are cooked through when internal temperature registers at 165˚F.
Remove chicken breasts from the skillet and set aside on a plate.
Add ½ tablespoon oil to the skillet if needed, and heat over medium-high heat.
Add diced onions to the heated oil and cook for 1 minute, or just until starting to soften.
Stir in garlic and cook for 10 seconds.
Add green beans and capers to the skillet; season with ¼ teaspoon salt and cook for 4 to 5 minutes, or until green beans are crisp tender. Stir occasionally.
Stir the previously prepared wine sauce and add it to the skillet. Cook for 10 seconds.
Return chicken and its juices back to the skillet; turn chicken to coat with the sauce.
Cook for a minute or two, or until sauce is a little thickened.
Remove from heat.
Garnish with parsley. Serve.