In a bowl, whisk together olive oil, balsamic vinegar, Dijon mustard, minced garlic, oregano, salt, and black pepper to create the marinade.
Add the chicken breasts to the marinade, ensuring they are well coated. Cover and refrigerate for at least 30 minutes (or up to 4 hours for more flavor).
Preheat the grill or grill pan over medium-high heat.
Remove the chicken from the marinade and shake off any excess. Discard the remaining marinade.
Grill the chicken breasts for about 6-7 minutes on each side, or until the internal temperature reaches 165°F (75°C). If using cheese, place a slice on top of each breast during the last minute of grilling to melt.
While the chicken is grilling, toast the hamburger buns on the grill or in a toaster.
Once the chicken is cooked, let it rest for a minute. Slice the chicken if desired.
Assemble the sandwiches by placing a grilled chicken breast on each bun. Top with lettuce, tomato slices, red onion, and your choice of condiments. Place the top half of the bun on each sandwich.
Serve immediately and enjoy!