How to Make It
Light a grill. When the fire is medium hot, brush the grate with oil. Skewer the garlic cloves on a 4-inch bamboo skewer. Grill the chiles and garlic, turning occasionally, until blackened all over, about 6 minutes for the garlic and 15 to 20 minutes for the chiles. Transfer the chiles and garlic to a medium bowl, cover with plastic wrap and let cool. Peel, core and seed the chiles and cut them into 1/4-inch strips. Peel the garlic cloves.
Transfer the garlic and two-thirds of the chiles to a blender. Add the cilantro, lime juice, cumin, oregano and water and puree until smooth. Season the sauce with salt and pepper and transfer to a small saucepan. Simmer over moderate heat, stirring occasionally, until the sauce is slightly thickened, about 5 minutes. In a small bowl, combine the remaining roasted chiles with the cheese.
Shape the meat into 4 patties about 3/4 inch thick and season very generously with salt and pepper. Grill the burgers over a medium-hot fire for about 10 minutes, turning once, for medium-rare meat. Move the burgers away from the heat. Brush the cut sides of the buns with oil and grill until lightly toasted, about 1 minute.&
Set the grilled burgers on the bottom halves of the buns. Top the burgers with the chile-cheese mixture and a generous spoonful of the salsa verde. Cover the green-chile burgers with the buns and serve right away, passing the remaining salsa verde on the side.