For the marinade:In a large bowl or resealable plastic bag, whisk together 5 Tbsp. olive oil and remaining marinade ingredients. Add chicken thighs and toss to evenly coat with the mixture. Refrigerate and let marinade for at least 1 hour, or overnight.
For the salad:When ready to cook, set your grill pan or outdoor gas grill to medium-high heat. Brush with 1 teaspoon olive oil. Once grates are hot, grill chicken (about 10 minutes each side) until golden and cooked through (juices should run clear; grill in batches, if needed). Once chicken is cooked, transfer to a cutting board and let rest.
Return your grill to medium-high heat. Add 1 teaspoon olive oil. Grill bell pepper strips and onions until lightly charred, or to your desired doneness (about 3-5 minutes).
Slice chicken into strips and lay on top of individual bowls of lettuce. Top with grilled bell peppers, onions, and avocado slices. Sprinkle each bowl with crumbled queso fresco. Add fresh cilantro and lime wedges, if desired.