Heat grill to high. Brush chicken and onions on both sides with oil and season with salt and pepper. Grill chicken for 4 minutes on each side or until slightly charred and just cooked through. Remove to a plate, loosely tent with foil and let rest 5 minutes before thinly slicing. Grill onions for 3 to 4 minutes per side or until slightly charred and just cooked through. Place the chicken and onion rings in a large bowl and add the carrots, parsley, and scallions.
Place the mayonnaise, broth, vinegar, garlic, paprika, salt, and pepper in a food processor and process until smooth. Pour the mayonnaise mixture over the chicken and gently toss to combine. Transfer the mixture to a platter, top with vegetable chip sticks, and serve.