Heat oil in a 10" skillet over medium-high heat. Add plantain halves and fry, turning once, until light brown, about 3 minutes. Transfer to a cutting board and split each plantain in half lengthwise, keeping them hinged on the inward-curved side. Place on cutting board cut sides down and, using a plate, flatten each into a 1/4″-thick round disk. One at a time, return each plantain disk to the oil and fry, turning once, until crisp and golden brown, about 4 minutes. Using tongs, transfer to paper towels to drain.
Purée mayonnaise, cilantro, and avocado in a food processor until smooth; season with salt and pepper. Transfer to a medium bowl and add chicken; toss to coat. Place chicken mixture on one plantain disk and top with lettuce and tomato. Cover with remaining disk.