Brush the chiles with oil and season with salt and pepper. Grill until blistered on all sides and soft, about 12 minutes. Remove the chiles from the grill, place in a bowl, cover with plastic wrap and let sit 15 minutes. Peel, stem and seed the chiles. Slice the chiles and set aside.
Make 4 loosely-packed burger patties by tossing some of the meat from hand to hand. press your thumb into the center of each patty to make a little well. (At this point, you can chill or even freeze the burgers briefly so that they hold their shapes better when cooked.) Brush with oil and season with salt and pepper. Grill until slightly charred on both sides and cooked to medium doneness, about 8 minutes. Top each burger with 2 slices of cheese, close the cover of the grill and continue cooking for about 30 seconds or until the cheese just melts. Put the buns on the grill, cut side down and grill until slightly charred, about 15 seconds. Place the burgers on the bottom half, put a few slices of the poblano on top of the cheese and cover with the burger tops.