Place the chicken breasts in a mixing bowl and add 1 tablespoon of balsamic vinegar, 1 tablespoon of olive oil or avocado oil, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Stir to coat the chicken thoroughly in the marinade. Cover and refrigerate the bowl while you prepare the green beans. I like to let the chicken marinate for at least 10 minutes for extra flavor.
Trim the pointy ends off of all the green beans and place them in another mixing bowl. Add 1 tablespoon of olive oil or avocado oil, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Toss the beans until they are fully coated. Hold off on adding the balsamic vinegar for now; it will be added during cooking.
Heat a non-stick pan over medium-high heat. Once the pan is hot, add the prepared green beans from Step 2. Cook, stirring occasionally, for about 5 minutes or until the beans begin to brown slightly on the edges. Add 2 tablespoons of balsamic vinegar, then reduce the heat to medium-low and cook for 2 more minutes. Transfer the cooked green beans to a plate or bowl while you cook the chicken. For a bolder flavor, I sometimes add an extra splash of balsamic vinegar at the end of cooking.
Return the same non-stick pan to medium heat. Place the marinated chicken breasts from Step 1 onto the pre-heated pan and cover. Cook for 6 minutes. Flip the chicken breasts, cover again, and cook for another 6 minutes. Check the chicken for doneness by slicing into the thickest part to ensure it's fully cooked. I like to let the chicken rest for a couple of minutes off the heat to keep it juicy.
Lower the heat to medium-low and return the cooked green beans from Step 3 to the pan with the cooked chicken from Step 4. Cook everything together for a few more minutes, stirring gently until heated through. Serve immediately. Don't forget to double check that the chicken is cooked all the way through before serving!