Combine shallot and lime juice in a large jar or medium bowl; season with sea salt and pepper and let sit 10 minutes. Add sesame oil, cooking oil, soy sauce, and honey or maple syrup to shallot mixture and whisk well. Season with salt and pepper, to taste.
Position an oven rack 6 inches from the broiler. Pour ¼ cup water into a rimmed 17 x 12-inch baking pan, and fit with a wire rack.
Season chicken wings with sea salt and pepper; then toss with some of the marinade so they are nicely coated. Arrange in a single layer on the rack. Prick the chicken skin with the tip of a knife. Broil for about 10 minutes, turning once.
Meanwhile, heat a drizzle of oil in a large cast-iron--or other--heavy skillet over medium-high heat. Cook peppers, turning occasionally, until they begin to blister on all sides. Remove from pan and set aside.
Remove wings from oven and toss with all but a ¼ cup of the marinade. Place back on the wrack and broil for about another 10 minutes until wings are cooked through and slightly charred.
Transfer wings to a platter. Scatter with shishitos and fresh basil and drizzle with the remaining marinade.