A peachy, peppery relish makes a lively contrast to the grilled chicken in these fun sandwiches. Make it a meal: Dress a sliced tomato and cucumber salad with balsamic vinegar and extra-virgin olive oil to serve alongside.
Heat 1 teaspoon of the oil in a large nonstick skillet over medium-low heat. Add onions and cook, stirring, until softened and starting to color, 5 to 7 minutes. Add sugar, stirring until it dissolves and starts to bubble, about 2 minutes. Add peaches and cook until the mixture turns golden brown, about 4 more minutes. Add vinegar and sherry, bring to a simmer and cook, stirring, until thickened and jamlike, 5 to 10 minutes. Stir in crushed peppercorns and season with salt. Transfer to a bowl and set aside.
Prepare grill or preheat broiler.
Place each chicken breast between two layers of plastic wrap and flatten gently with a rolling pin or heavy skillet until approximately ¼ inch thick. Brush chicken breasts lightly with the remaining 1 teaspoon oil and season with salt and pepper. Grill or broil the chicken until no longer pink inside, 3 to 4 minutes per side. While the chicken is cooking, toast the rolls on the grill or under the broiler.
Place lettuce leaves on the bottom halves of the toasted rolls, followed by the chicken. Top with the peach-onion relish and the roll tops.
Tip: To peel peaches: Bring a medium saucepan of water to a boil. Immerse peaches in boiling water for 1 minute. Remove with a slotted spoon and plunge into a bowl of ice water to cool slightly. Transfer to a cutting board. Remove the skins with a paring knife.
Note: Sherry is a type of fortified wine originally from southern Spain. Don't use the “cooking sherry” sold in many supermarkets—it can be surprisingly high in sodium. Instead, purchase dry sherry that's sold with other fortified wines in your wine or liquor store.