Ingredients

1. Olive oil, for brushing
2. 4 unpeeled garlic cloves
3. 1 1/2 pounds New Mexico or poblano chiles
4. 2 tablespoons finely chopped cilantro
5. 2 teaspoons fresh lime juice
6. 1/2 teaspoon ground cumin
7. 1/2 teaspoon dried oregano, crumbled
8. 1/4 cup water
9. Kosher salt and freshly ground pepper
10. 4 ounces sharp white Cheddar cheese, shredded (about 1 cup)
11. 1 1/2 pounds ground sirloin, at room temperature
12. 4 hamburger buns

Directions

1. 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. 
2. 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. 
3. 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. 
4. 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.

Notes

Beer Recommendation: Set off the earthy chiles and the spicy salsa that top the burgers with the soft, smooth, gently fruity Sante Fe Pale Ale from the Santa Fe Brewing Company in New Mexico.