Preheat the oven to 450 degrees F.
Season the chicken breast on both sides with salt and pepper. Heat a large saute pan over high heat. When the pan is hot, add the oil and heat to almost smoking. Add the chicken, skin side down and cook for 4 minutes. The skin should be golden brown. Turn the chicken over and place in the oven. Roast until the chicken is just cooked through and the juices run clear, about 15 to 20 minutes.
While the chicken is in the oven, prepare the greens. Fill a very large bowl halfway full with ice water. Quarter the Napa cabbage lengthwise and remove the core. Thinly slice the cabbage widthwise and place in the ice water. Repeat this same process for the radicchio. Halve the romaine lettuce lengthwise and then thinly slice. Add to the water. Mix all of the greens well in the ice water and let sit for about 5 minutes. Drain. Place in a clean kitchen towel and dry well. Be careful not to bruise the greens. Refrigerate.
Remove the chicken from the oven and cool enough to handle. Remove the skin and slice the chicken. Discard the skin and bones. Roughly chop the sliced chicken and place in a large bowl. The chicken should still be warm. Add some of the vinaigrette and toss well. Add the greens and a little more vinaigrette and toss until everything is well coated.
Place a small mound of greens on 4 plates. Top each mound with a spring roll basket and lightly press to stabilize. Fill the baskets with more of the salad. Top with some of the fried springroll strips, a sprig of cilantro, peanuts, and a lime wedge.
Heat a deep fryer to 375 degrees F. Or fill a deep pot halfway full with oil and heat to 375 degrees F.
Roll up 2 of the egg roll wrappers and julienne. Set aside. Place 1 of the remaining wrappers in the hot oil for 5 seconds to soften. Then, carefully push the center of the wrapper down with a ladle. Fry just until the oil stops sizzling, about 15 to 30 seconds. The basket should be golden brown. Remove with tongs to a paper towel lined plate to drain. Repeat with the remaining 3 wrappers. Once all of the baskets are made, add the julienne strips and fry just until the oil stops sizzling and the strips are golden brown, about 30 seconds. Remove with a slotted spoon and drain on paper towels.
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