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Preheat the barbecue on high, then reduce heat to medium. Grill chicken breasts about 20 minutes, turning and brushing with marinade, until they are fully cooked and no longer pink.
Transfer to a plate and cover loosely with aluminum foil.
For the remoulade, combine mustard and mayonnaise in a bowl. Add vegetables, herbs, salt, and pepper and stir together. (This remoulade can be made with a wide variety of ingredients so feel free to get creative!)
Place two slices of cheese in each hotdog bun and toast on the grill.
Slice chicken into strips and divide up between hotdog buns.
Top with remoulade and enjoy this fresh take on a chicken sandwich!