1.Preheat the oven to 375 degrees F
2.Put the chile on a rimmed baking sheet, rub with 1 tablespoon of the oil, and season with salt and pepper. Roast in the oven until the skin of the chile is blackened, about 15 minutes. Remove the chile from the oven, place in a bowl, cover with plastic wrap, and let the chile steam for 15 minutes. Peel, stem, and seed the chile and then finely dice it.
3.To make the queso sauce, melt the butter in a small saucepan over medium heat. Whisk in the flour and cook for 1 minute. Add the milk, increase the heat to high, and cook, whisking constantly, until slightly thickened, about 5 minutes. Remove from the heat, add the cheese and whisk until melted. Season with salt and pepper and keep warm.
4.Divide the meat into 4 equal portions (about 6 ounces each). Form each portion loosely into a 3/4-inch-thick burger and make a deep depression in the center with your thumb. Season both sides of each burger with salt and pepper.
Heat a gas grill to high. Brush the burgers with remaining 2 tablespoons of canola oil and grill until desired doneness is reached.
5.Place the burgers on the bun bottoms and top each with a few tablespoons of queso sauce, some of the poblano, and the chips. Cover with the bun tops and serve immediately.