
Preheat grill or grill pan to high. Pound or butterfly cutlets to an even thickness. Sprinkle with 1/4 teaspoon salt and pepper. Grease the grill rack or pan with a piece of oil-dampened paper towel.
Grill the chicken, turning once, for 10 minutes. Brush the chicken with 2 tablespoons apricot preserves and continue grilling until an instant-read thermometer inserted into the thickest part registers 145°F, 2 to 5 minutes more. Grill apricot or nectarine halves on the cooler part of the grill, turning occasionally, until tender and have developed grill marks, about 4 minutes. Transfer the chicken and apricots to a clean cutting board and let rest for 5 minutes.
Meanwhile, whisk the remaining 1 tablespoon preserves, vinegar, shallot, oil, 1/4 teaspoon salt and pepper to taste in a large bowl.
Cut the fruit into wedges and the chicken into slices. Place the lettuce in a large bowl and add the sliced fruit and chicken. Add the nuts. Add the dressing and toss to coat. Serve immediately.
