Light your grill and set it up for two zone cooking, or preheat your oven to 325 °F.
While your grill or oven is preheating, prepare the sliders. Leaving the rolls connected, separate the tops from the bottoms by slicing them in half horizontally. Place the bottom half of the rolls in a 9×13 inch baking dish and cover with two cups of shredded cheese.
In a medium size mixing bowl, mix together chicken, enchilada sauce and green chilis. Add shredded chicken mixture on top of the cheese, then top with another two cups of cheese.
Place the tops of the rolls on the sliders and put them on the grill over indirect heat. Allow them to bake at 325 °F for 5 minutes.
While sliders are baking, melt the butter in a small saucepan over low heat. Then, whisk in the Mexican oregano and garlic.
Open the grill or oven and, using a pastry brush, brush half of the warm butter glaze on top of the rolls until uniformly covered. Close the lid and allow the sliders to continue to bake for an additional 15 minutes or until golden brown and the cheese is melted.
Remove sliders from the grill or oven, and brush on remaining butter glaze. Garnish with cilantro and serve hot.