Pat the chicken breasts dry with paper towels and season both sides with salt and pepper.
Melt the butter with the olive oil in a large skillet over medium-high heat.
When the oil is hot, add the seasoned chicken and cook undisturbed until golden brown on the bottom, about 6 to 7 minutes. Flip and cook an additional 6 to 7 minutes, or until the chicken is nicely browned and reaches an internal temperature of 165°F.
Transfer the cooked chicken to a plate and tent loosely with foil to keep warm.
In the same pan used to cook the chicken, add the chicken broth, balsamic vinegar, lemon juice, garlic, and honey. Use a wooden spoon to scrape up the browned bits from the bottom of the pan. Simmer, stirring occasionally, until the sauce has thickened enough to coat the back of a spoon, about 5 minutes.
Slice the chicken and spoon the sauce overtop. Serve immediately.