Heat a large cast-iron skillet over high heat. Add the tomatillos, jalapeno and red onion chunks. Cook, turning, until blistered, 5 to 6 minutes for the onion and 7 to 8 minutes for the tomatillos and jalapeno. Transfer to a cutting board; cool slightly. Chop the tomatillos and onion; seed and chop the jalapeno.
Heat 1 tablespoon olive oil in a medium skillet over medium-high heat. Add the tomatillos, chopped onions and half of the jalapeno. Cook until the vegetables release their juices, then add the chicken, the juice of 1 lime and 1/2 teaspoon salt. Simmer until warmed through, about 2 minutes. Keep warm over low heat.
Toss the watermelon, 1/2 cup cilantro, the sliced red onion, the remaining jalapeno, juice of 1 lime, 1 tablespoon olive oil and half of the cheese in a bowl. Season with 1/4 teaspoon salt and a few grinds of pepper. Warm the tortillas in a dry skillet.
Add the remaining 1 cup cilantro to the chicken mixture. Divide among the tortillas; top with the watermelon salad, the remaining cheese and the avocado. Serve with lime wedges.