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1
Preheat the oven to 375 degrees F.
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2
Put the chile on a rimmed baking sheet, rub with 1 tablespoon of the oil, and season with salt and pepper.
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3
Roast in the oven until the skin of the chile is blackened, about 15 minutes.
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4
Remove the chile from the oven, place in a bowl, cover with plastic wrap, and let the chile steam for 15 minutes.
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5
Peel, stem, and seed the chile and then coarsely chop it.
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6
To make the queso sauce, melt the butter in a small saucepan over medium heat.
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7
Whisk in the flour and cook for 1 minute.
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8
Add the milk, increase the heat to high, and cook, whisking constantly, until slightly thickened, about 5 minutes.
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9
Remove from the heat and whisk in the cheese until melted; season with salt and pepper.
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10
Keep warm.
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11
Divide the meat into 4 equal portions (about 6 ounces each).
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12
Form each portion loosely into a 3/4-inch-thick burger and make a deep depression in the center with your thumb.
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13
Season both sides of each burger with salt and pepper.
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14
Cook the burgers, using the remaining 1 1/2 tablespoons oil (see page 16).
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15
Place the burgers on the bun bottoms and top each with a few tablespoons of queso sauce, chips, and some of the poblano.
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16
Cover with the bun tops and serve immediately.
