Prepare the chicken: Preheat a grill to medium on one side. Combine the hoisin sauce, scallions, garlic, ginger, jalapeno, lime zest and juice, vinegar, and salt and pepper to taste in a mini food processor or blender; process until almost smooth. Season the chicken with salt; toss with the hoisin mixture in a large bowl.
Make the salad: Bring the vinegar, sugar and 1 1/2 teaspoons salt to a boil in a small saucepan, stirring. Remove from the heat; stir in 1/4 cup ice cubes until melted. Toss the cucumber and red onion in a serving bowl with the vinegar mixture; place in the refrigerator.
Grill the chicken over direct heat until the skin is crisp, 5 minutes per side, basting with extra marinade. Transfer to the cooler side of the grill; cover and grill until cooked through, 10 to 15 more minutes. Serve with the cucumber salad and lime wedges.