In a small bowl whisk together the lime juice, honey, shallot, mustard, rosemary, salt, and pepper. Slowly whisk in the oil to make a smooth dressing.
Put the chicken breasts into a large bowl, add 1/4 cup 60 milliliters of the dressing, and turn the breasts to coat well. Let marinate at room temperature for 15 to 30 minutes before grilling. Put the onion slices into a medium bowl, spoon 2 tablespoons of the dressing over the top, and turn gently to coat evenly.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Prepare the grill for cooking over medium heat (400°F).
Grill the chicken, smooth (skin) side down first, over direct medium heat, with the lid closed, until an instant-read thermometer inserted into the thickest part the chicken registers 165°F, 8 to 12 minutes, turning once or twice. At the same time, grill the onion slices over direct medium heat until charred in spots and tender, about 8 minutes, turning once. Remove the chicken and onion from the grill and let the chicken rest for 3 to 5 minutes.
Grill the chicken, smooth (skin) side down first, over medium heat, with the lid closed, until an instant-read thermometer inserted into the thickest part the chicken registers 165°F, 8 to 12 minutes, turning once or twice. At the same time, grill the onion slices over medium heat until charred in spots and tender, about 8 minutes, turning once. Remove the chicken and onion from the grill and let the chicken rest for 3 to 5 minutes.
In a large serving bowl combine the lettuce, apples, grapes, cashews, and cheddar. Cut the chicken into thin strips and roughly chop the onion. Add the chicken and onion to the bowl. Drizzle with enough of the remaining dressing to coat lightly (you may not need all of it) and mix well. Serve at room temperature.
