Preheat oven to 375 degrees, with rack in center. Between two sheets of waxed paper, pound chicken breasts with a meat pounder until about 1/2 inch thick. Transfer chicken to a resealable plastic bag. Add lime juice, oil, salt, and pepper; refrigerate.
Place tortillas on a baking sheet. Brush tops with oil, and season with salt and pepper. Bake until crisp and golden, 10 to 12 minutes. Remove from oven; set aside to cool.
Heat a grill or grill pan over medium-high heat. Remove the chicken from refrigerator; discard marinade. Grill chicken until browned and cooked through, about 5 minutes per side. Let rest until cool enough to handle, about 5 minutes; cut into strips.
Brush onions with oil; grill until browned, about 3 minutes per side. Serve the tortillas topped with lettuce, tomatoes, avocados, chicken, onions, cilantro, and sour cream.