Gently form the ground chicken into four patties of equal size, 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 the patties to prevent them from forming a dome as they cook. Refrigerate the patties until ready to grill.
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, season with salt and pepper, and then grill over direct medium heat, with the lid closed, 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, until fully cooked but still juicy (165°F), 8 to 10 minutes, turning once. During the last minute of grilling time, place a slice of cheese on top of each patty to melt. Transfer the patties to a platter.
Brush the bread on one side with oil and toast the slices over direct medium heat until browned on both sides, about 1 minute, turning once. Serve the burgers on rye with the mayonnaise mixture and some onions on top.
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