Preheat the oven to 375 degrees F.
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 coarsely chop it.
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 and whisk in the cheese until melted; season with salt and pepper. Keep warm.
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. Cook the burgers, using the remaining 1 1/2 tablespoons oil. See how Bobby cooks perfect burgers
Place the burgers on the bun bottoms and top each with a few tablespoons of queso sauce, chips, and some of the poblano. Cover with the bun tops and serve immediately.
Toasted Burger Buns
I think the taste and the texture of buns are best when lightly toasted. To toast a bun on a grill, grill pan, or griddle, split the bun open, place it cut side down on the grill, and grill until light golden brown, about 10 seconds. Alternatively, you can place the split bun halves cut side up on a baking sheet and cook them under a preheated broiler until light golden brown, about 30 seconds. Keep a close eye on them and don't let them get too brown or else that texture will take on a life of its own and defeat the purpose of starting with a soft bun.