In a small bowl, mix the brown sugar, fennel seeds, paprika, garlic powder, pepper, and salt.
In a medium bowl, combine the tomatoes, onion, olives, capers, basil, garlic, olive oil, lemon juice, lemon zest, and 1/4 teaspoon each kosher salt and pepper. Let stand while the meat grills. Before serving, adjust the lemon juice, salt, and pepper to taste.
Build a hot charcoal fire or heat a gas grill with all burners on high for at least 10 minutes. While the grill heats, lightly coat both sides of the pork chops, chicken breasts, or turkey cutlets with the oil and rub with the spice rub. Clean the hot grate with a wire brush; oil the grate.
Grill the meat (uncovered for charcoal; covered for gas) over direct heat on the hottest part of the grill, taking care not to crowd the meat. Cook until the meat forms impressive grill marks on one side, 2 to 3 minutes. Turn and continue to grill (uncovered for charcoal; covered for gas) until the meat is just firm to touch and just cooked through, 1 to 2 minutes for turkey cutlets; 3 to 4 minutes for pork chops and chicken breasts, depending on their thickness. (If grilling pork, watch out for flare-ups.) Transfer to a serving platter and let rest for 5 minutes. Serve with the salsa spooned alongside or over the meat.
Whichever meat you choose, a fruity, dry rosé like Bonny Doon Ca del Solo Big House Pink would be a delicious match.