Place the ground chicken in a bowl with the scallions, parsley, dill and paprika; season with salt and pepper. Combine the mixture with a fork and form into four 1-inch-thick patties.
In a heavy nonstick skillet, heat the EVOO, 1 turn of the pan, over medium-high heat. When the oil is hot, add the burgers and cook for 6 minutes on each side.
Meanwhile, preheat the broiler. Place the red peppers, sour cream and shallot in a food processor. Process until smooth, about 20 seconds. Transfer to a small dish.
Toast the rolls under the broiler and spread lightly with softened butter. Slather the rolls with the red pepper sauce. Set a burger on each bun bottom and top with the radish and cucumber slices, 2 lettuce leaves and the roll top.