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1
Roast the chiles until they are blackened and blistered on all sides, 5 to 7 minutes: You can do this in the broiler or by roasting over a gas burner on medium-high heat.
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2
Transfer to a large heatproof bowl and cover the bowl tightly with plastic wrap. Let the chiles steam until they're cool enough to handle, about 15 minutes. Using your hands or a butter knife, scrape away and discard the chile skins carefully, as to not tear the chiles.
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3
Cut a slit lengthwise (from stem to bottom) of each pepper and remove the seeds and veins. Season the inside and outside of each chile with salt and pepper, then stuff each with about 1/4 of the cheese.
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4
In a medium bowl, whisk the egg yolks until lightened in color and frothy, about 2 minutes. Place the egg whites and salt in a clean, dry bowl of a stand mixer with the whisk attachment and beat on high speed until you reach stiff peaks, about 1 1/2 minutes.
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5
Add the egg yolks and 1/3 cup flour to the egg whites and fold with a rubber spatula until just combined, taking care to not deflate the egg whites.
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6
In a heavy-bottomed Dutch oven, heat 2 inches of oil to 360u00b0 F. Season the remaining 1/4 cup of flour with salt, pepper, and paprika. Dredge the stuffed chilies in the flour mixture, shaking off any excess. Then dip each chile into the egg batter, letting excess drip off. If the chile is fragile, you might want to use a spoon to move the mixture over top. Place carefully in the hot oil (it helps to use a large fish spatula to slide the chile in) and fry until golden, about 5 minutes per side. Remove to a wire rack or a paper-towel lined baking sheet.
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7
To make the tortilla chips, spread a few handfuls of the chips out on a baking sheet and sprinkle shredded Monterey Jack cheese over top. Send into the oven (under the broiler, if you're careful) until the cheese has melted. Let cool.
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8
If you're looking for recipes for the guacamole, bean spread, and salsa, we like My Favorite Black Bean Dip, Roberto Santibanez' Classic Guacamole, and Not Recipe Salsa.
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9
To assemble the sandwiches, spread the beans over one side of the bread. Place a poblano on top, then balance the cheesy chips (break them up if you want) over top. Blob with guacamole and salsa. Top with cilantro leaves.
