Pat the chicken dry and slice each breast into two thinner cutlets.
Whisk the olive oil, lemon juice, garlic, salt, thyme, and pepper together in a large bowl. Add the chicken and toss to coat.
Cover and marinate in the refrigerator for 2 hours and up to 24 hours.
In a small bowl, stir the mayonnaise, barbecue sauce, and dijon mustard together. Refrigerate until ready to use.
Preheat a grill or grill pan over medium-high heat. Brush the grill with oil.
Grill the chicken until cooked through, about 3 minutes per side. During the last minute of cooking, add a slice of cheese to each piece of chicken. Cover and cook until the cheese is melted. Remove from the grill and set aside.
Brush the grill with more oil and grill the onion rings until grill marks appear and the onions are cooked through, about 4 minutes per side.
Assemble the sandwiches: spread each side of the bun with the sauce. Top with lettuce, grilled chicken, bacon and grilled onions. Serve immediately.