In a small bowl combine the rub ingredients.
Remove the tenders from the underside of each breast (save for another use). One at a time, place each breast, smooth side down, between 2 sheets of plastic wrap and pound to an even ¼-inch thickness. Lightly brush the chicken with oil and season both sides with the rub.
Prepare the grill for direct cooking over high heat (450° to 550°F).
In a large bowl combine the relish ingredients, including salt to taste.
Brush the cooking grates clean. Grill the chicken, smooth side down, over direct high heat, with the lid closed as much as possible, until no longer pink, 3 to 4 minutes. Turn over and grill just to sear the surface, about 1 minute. Transfer the chicken, with the first grilled sides facing up, to a serving platter or individual plates. Spoon the relish over each piece and squeeze fresh lemon juice on top just before serving.
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