Set the chicken breast halves one at a time on parchment or wax paper
Cover with a second sheet of paper and pound with a rolling pin or large heavy pan until the chicken is about 1/2" thick. Repeat with the other pieces of chicken, changing the paper as needed
Sprinkle each flattened piece of chicken with salt and pepper on both sides
Add the flour to a plate. If you are gluten intolerant, use a gluten free flour blend like Bob's Red Mill
Dredge each piece of chicken in the flour on both sides
Heat the butter and olive oil in a large sauté pan
Heat the pan over medium-high heat until the oil and butter are bubbling, being careful not to let it burn
Add the chicken pieces to the pan and sauté giving them room to cook. You may have to cook them in batches
Sauté the chicken until it is a lovely golden brown and cooked through 3 - 4 minutes per side (or more if your pieces are thicker)
Remove the chicken from the pan to a plate and cover with foil to keep warm
To make the sauce - add the 1/2 cup of balsamic vinegar to the pan and boil, scraping up the brown bits in the pan with a spoon, until the vinegar has reduced by half (See the notes below for additional sauce ideas)
Once the vinegar has reduced, remove from the heat, taste, and adjust seasonings if needed
Place the chicken on top of a salad or your favorite side dish, drizzle with the sauce and enjoy!