Herb-marinated chicken paillards—breast fillets butterflied and flattened until very thin—grill up quickly, so stick close by so they don’t overcook. Halved ripe tomatoes and tomatillos still in their husks need just a few minutes on a hot grill to give them a nice roasted flavor.
Carefully slice open each chicken breast horizontally so they can be opened up like a book. Place between two pieces of plastic wrap and lightly pound with a rolling pin or flat side of a meat mallet until very thin. Place in a shallow dish.
Mix 3 tablespoons oil, lime juice, 3 cloves garlic, rosemary, chili powder, and cumin in small bowl. Pour over the chicken, being sure they are coated with marinade. Cover and refrigerate until ready to grill.
Heat grill to medium-high heat. Brush cut sides of tomatoes with remaining 1 tablespoon oil; season with salt and pepper. Place chicken on grill; cook 5 to 7 minutes, turning once, until no longer pink in center. Place tomatoes, cut-side down, around edge of grill. Place tomatillos, still in their papery husks, on grill alongside tomatoes. Cook, turning a few times, until husks are browned.
To make salsa, pull husks off tomatillos. Finely chop; mix with remaining salsa ingredients. Serve with grilled chicken and tomatoes.
Boneless skinless thighs may be substituted for boneless skinless breasts.
Nutritional values may change if you choose to substitute another chicken product. If substituting frozen chicken for this recipe, thaw before cooking. If substituting fresh whole boneless skinless chicken breasts, cut the whole breast in half before preparing.
You can either hand-chop the salsa or buzz it up in the food processor—stirring the avocado in at the end.