Preheat the oven to 400 degrees F. Puree the mayonnaise, parsley, scallion, garlic, 2 tablespoons water, 1 tablespoon lemon juice, 1/2 teaspoon salt and 1/4 teaspoon pepper in a food processor; transfer to a bowl.
Add half of the shrimp to the food processor and pulse until a thick paste forms. Add the remaining shrimp, 1/2 cup panko, 2 tablespoons of the prepared dressing, the egg and lemon zest. Pulse until combined but still chunky. With oiled hands, shape the mixture into eight 1/2-inch-thick patties. Transfer to a baking sheet; freeze 5 minutes.
Put the remaining 1/2 cup panko on a plate. Heat the vegetable oil in a large nonstick ovenproof skillet over medium heat. Coat the patties on both sides with the panko; cook until golden, 2 minutes per side. Transfer the skillet to the oven and bake until the patties are cooked through, 5 to 7 minutes.
Meanwhile, toss the greens, radishes, 2 tablespoons of the dressing and the remaining 1 tablespoon lemon juice; season with salt and pepper. Serve the shrimp cakes with the salad and remaining dressing.