Combine scallion greens, radish tops, lemon juice, lemon zest, garlic, nuts, sea salt and black pepper in the bowl of a food processor. Pulse 20 times; then scrape the bowl with a spatula into a mound. Add olive oil in a steady stream and continue to pulse until pesto reaches desired consistency. Adjust the salt and lemon, to taste, and set aside.
Season chicken generously with sea salt and pepper and place in a large pan. Pour ¼ cup vinegar over chicken and let sit 20-30 minutes.
Preheat oven to 450 degrees. Remove chicken from marinade, pat dry; then rub skin with a large drizzle of olive oil. Place chicken, skin-side up, in a large cast iron skillet. Roast in oven until juices run clear and meat is no longer pink, 35-40 minutes. If you have a meat thermometer, insert it into thickest part of thigh to gauge doneness—it should read 165 degrees. Turn oven to broil and allow chicken skin to crisp, 1-2 minutes. Remove from skillet and let rest.
Heat butter in same skillet over medium-high. Add radishes, season with sea salt and pepper, and cook, stirring occasionally, until radishes are browned and tender, about 5 minutes. Add treviso and toss to coat; continue to cook until just wilted, about 2 minutes. Add remaining tablespoon of vinegar; toss to combine.
Insert a knife between thigh and breast, gently pulling leg away to separate chicken into two pieces. Serve over treviso and radishes along with scallion radish top pesto.