Shape the ground chicken into four equal-sized patties, each about ¾ inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of each patty so the centers are about ½ inch thick. This will help the patties cook evenly and prevent them from puffing on the grill. Cover the patties with plastic wrap and refrigerate until you are ready to grill them, or for at least 5 minutes.
In a small bowl mix the mayonnaise and mustard.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Lightly coat the onion slices on both sides with oil and season with salt and pepper. Brush the cooking grates clean. Grill over direct medium heat, with the lid closed as much as possible, until tender, 6 to 8 minutes, turning once. Remove from the grill.
Lightly brush the patties on both sides with oil and season evenly with salt and pepper. Grill over direct medium heat, with the lid closed as much as possible, until fully cooked but still juicy, 12 to 14 minutes, turning once. During the last 2 to 3 minutes of grilling time, place a slice of cheese on top of each patty. Transfer the patties to a platter.
Brush the bread on one side with some oil and toast the slices over direct medium heat until browned on both sides, 1 to 2 minutes, turning once. Serve the burgers on rye with the mayonnaise mixture and some onions on top.
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