Set a wide-mouthed 3-quart saucepan with lid over medium-high heat and add the oil to the pan. Season the chicken pieces with the chili powder and salt and toss the evenly coat. Add the chicken to the pan and sear for 3 minutes per side. Add the onions to the pan and saute for another 3 minutes. Add the garlic to the pan and saute for 1 minute. Pour the chicken stock over the chicken and bring to a boil. Reduce to a low heat and cover. Continue to cook the chicken until very tender, about 1 hour, stir the chicken to shred it and continue to cook, covered, for another 30 minutes. Once the chicken is cooked, stir and mash to shred the chicken meat and add the lime juice to the chicken. Set aside as you prepare the nachos.
For the Nachos:
In a medium saucepan, melt the butter over medium-high heat and whisk in the flour. Cook, stirring constantly, for 2 to 3 minutes. Do not allow the flour to color. While whisking constantly, add the milk in a steady stream. Add the green chiles and bring the sauce to a boil. Cook, stirring frequently, for about 5 minutes, or until the sauce is thickened and bubbly. Add the Cheddar, salt, and cayenne pepper and stir until the cheese is melted and the sauce is smooth. Cover and remove from the heat.
Preheat the oven to 400 degrees F.
In a large, ovenproof shallow casserole or on a large baking sheet, spread half of the tortilla chips in an even layer. Top with half of the chicken, and half of the Monterey Jack cheese. Top with the remaining tortilla chips and chicken. Pour the hot cheese sauce over the chicken, then evenly spread the remaining grated Monterey Jack over the top. Bake until the cheese is melted and bubbly and the nachos are heated through, 8 to 10 minutes. Garnish with the sliced green onions, sour cream, pickled jalapenos and cilantro. Serve with the Pico de Gallo alongside.
For the Pico de Gallo:
In the bowl of a food processor, combine all the ingredients and pulse until the salsa is of uniform consistency but still slightly chunky, about 7 pulses. Transfer to a bowl and set aside until ready to serve the nachos.