Heat a grill or grill pan to medium-high. Brush chicken with 1 tablespoon oil and season with salt, garlic powder and paprika; brush onion rings with the remaining oil.
Whisk mayonnaise, rice vinegar, lemon juice, horseradish, brown sugar and pepper in a bowl. Set aside 1/2 cup for the cooked chicken and keep the remainder separate for the rolls.
Grill chicken for 5 to 7 minutes per side until the centers reach 165°F. Grill onion rings alongside for 8 to 10 minutes, turning once, until tender with browned edges.
Rest chicken for 5 minutes, then slice thinly across the grain. Toss with the measured 1/2 cup white sauce; separate the cooked onions into short manageable strands.
Split the rolls and toast only the cut faces on the grill for 30 to 60 seconds. Spread with the reserved sauce, then divide chicken, grilled onion and drained pickle chips among them.
Close the subs and cut across the middle. Serve one per person while the chicken remains warm and the pickles are cool.