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Preheat oven to 350 degrees F. Brush 1 tablespoon oil over the tortillas (both sides) cut in half then cut into strips. Spread out on a large baking sheet and bake, stirring once until browned and crispy, about 12 minutes. Set aside. Turn oven onto broil.
Heat 1 tablespoon oil in a large skillet over medium-high heat. Swirl the oil in the bottom of the pan to coat. Add the chicken, spreading out over the surface of the pan. Sprinkle with ½ teaspoon salt and ¼ teaspoon pepper. Cook stirring once or twice until browned and just cooked through, 5 to 8 minutes. Lift chicken out of the pan and transfer to a plate. Cover with foil to keep warm.
Return the skillet to medium-high heat and add the remaining 1 tablespoon olive oil. Add the squash, scallions, cumin, chili powder and the remaining ¼ teaspoon salt and ¼ teaspoon pepper and cook, stirring often until the squash is starting to soften and the spices are browning and fragrant, 3 to 5 minutes. Add broth, lime juice and oregano and stir to combine. Add the cooked chicken and any accumulated juices to the skillet and bring to a simmer. Remove from the heat and stir ¼ cup cilantro and about half of the tortillas.
Top with the remaining tortillas, and cheddar. Broil until the cheese is melted. Top with the remaining ¼ cilantro and serve immediately.