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Special equipment: a deep fry thermometer
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Place the sliced red onions in a small stainless steel bowl.
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With a fine grater or zester, zest the skin of 1 lime over the onions.
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Cut the 4 whole limes in half cross wise and squeeze the juice over the onions.
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Add a pinch of salt and mix.
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Submerge the onions in the juice; use the squeezed limes as a weight to push down any onions sticking out of the juice.
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Cover with a tight-fitting lid and refrigerate.
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Marinate the onions in the juice for as long as possible, at least 15 minutes or up to 3 days.
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Pour the oil into a large heavy pot that has at least a 6-quart capacity.
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(There should be at least 2 inches of oil in the pot.)
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Clip a deep-frying thermometer to the side of the pot and be sure the thermometer is touching the bottom of the pot.
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Heat the oil to 400 degrees F over medium heat (do not ever cover the pot).
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Place a chicken thigh on a cutting board and cut into 4 to 5 even-sized chunks.
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Repeat until all thighs are cut into chunks.
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Arrange all the chunks on a microwave-safe plate and season both sides with 1/4 tablespoon of adobo powder, as well as pepper.
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Place the plate in the microwave and cook on high for 1 minute and 30 seconds.
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Remove the plate and turn each piece of chicken over.
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Microwave again on high until the chunks are just cooked through, another 1 to 2 minutes, depending on your microwave.
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Let the chicken cool to room temperature.
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Put the egg whites in a bowl and beat with a fork until just foamy, about 1 minute.
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In a separate wide and shallow bowl or cake pan, combine the remaining 2 tablespoons of adobo powder and paprika with the cornmeal and whole-wheat flour.
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Mix thoroughly.
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Submerge the chicken pieces in the egg whites.
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With a fork, toss to coat.
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One at a time, lift the chicken pieces from the egg whites, transfer them to the flour mixture, and coat each piece thoroughly.
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Let the chicken sit in one layer over the flour mixture.
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Wash and dry your hands thoroughly prior to assembling the salad.
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Place the iceberg chunks in a bowl and add the cherry tomatoes.
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Drain the lime juice from the onions and reserve.
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To the salad, add the onions, a dash of salt, and 2 tablespoons of the reserved lime juice.
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Toss to dress evenly.
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Divide the salad among four serving plates.
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Divide the avocado pieces equally among the salads.
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Fry the chicken in the oil in four separate batches until deep golden brown, about 12 seconds.
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Using metal tongs or a stainless steel spider basket, remove the chicken pieces from the oil and transfer them to a plate with either a wire rack on top or paper towels to drain off any excess oil.
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(Turn the stove off immediately after your last batch.)
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Arrange the cooked chicken evenly around the individual salads.
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Place 4 cilantro sprigs over each salad.
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Serve with lime wedges.