Bring a pot of water to a boil; add the tomatillos, jalapenos, onion, and garlic. Simmer for 10 to 15 minutes, until the tomatillos are soft. Drain and cool slightly, then put them in a blender. Add the cilantro, lime juice, and cumin. Puree for a few seconds to blend, and then pour in about a 1/4 cup of water and process to a coarse puree; taste and season with a generous pinch of salt. You should have about 2 cups of this salsa verde.
Make a roux by melting the butter over medium-low heat in a thick-bottomed saucepan. Just as the foam subsides, sprinkle in the flour, stirring constantly with a wooden spoon or whisk to prevent lumps. Cook for 2 to 3 minutes to remove the starchy taste from the flour; don't allow it to brown. Gradually whisk in the chicken stock and simmer for 8 minutes to thicken. Once you have a good base, fold in 2 cups of the shredded Jack cheese; mix until completely melted into a sauce. Stir in the prepared salsa verde until incorporated; remove the green chili cheese sauce from the heat.
Make a quick salsa by combining the cherry tomatoes, onions, jalapeno, cilantro, and lime juice in a bowl; season with salt and pepper, tossing to combine.
To build these awesome nachos: Preheat the oven to 350 degrees F. Get a very large oven-proof platter and cover it with a few handfuls of tortilla chips, follow with a portion of the shredded chicken, a coating of the cheese sauce, and a nice sprinkle of the remaining shredded jack. Make 3 or 4 layers of the nachos, depending on the size of the platter. Bake the nachos until they are all hot and gooey, about 5 to 10 minutes.
Spoon the tomato salsa over the top of the nachos and serve with the sour cream on the side.