Preheat a grill or grill pan to high. Add the poblano and grill until charred on all sides, 10 to 12 minutes. Place in a bowl, cover and let stand for 15 minutes. Peel, halve and seed the poblano, then coarsely chop.
Meanwhile, in a medium saucepan, melt the butter over medium heat. Whisk in the flour for 1 minute. Add the milk, increase the heat to medium-high and cook, whisking constantly, until thickened, about 5 minutes. Whisk in the cheese until melted; season with salt and pepper. Keep warm.
Shape the ground beef into 4 patties; season with salt and pepper. Grill until charred on both sides, about 8 minutes for medium.
Place the burgers on the bun bottoms and top each with some warm cheese sauce, chopped poblano and tortilla chips. Set the bun tops in place.