While you're gathering ingredients, start by making the yogurt sauce since it benefits from a few minutes to meld flavors. In a small bowl, combine the yogurt, 1 minced garlic clove, dried dill, salt, and lemon juice. Stir until smooth and set aside. This gives the flavors time to develop while you prepare everything else. Next, finely shred the cabbage into thin 1/8-inch strips and set aside. Mince the 3 garlic cloves for the chicken and have all your dried spices measured out and ready (chili flakes, black pepper, and dried mint).
Cut the chicken breasts into 1-inch cubes for even cooking. Heat the olive oil in a large skillet over medium-high heat until it shimmers. Add the cubed chicken and let it cook undisturbed for 2-3 minutes to develop a light golden color on the bottom. Add the minced garlic cloves, chili flakes, black pepper, and dried mint, stirring constantly for about 30 seconds to toast the spices and release their aromatics. I like to add the spices right after the chicken gets some color because it prevents them from burning while building deeper flavor.
Continue cooking the chicken, stirring occasionally, for 8-10 minutes until the chicken is cooked through (internal temperature should reach 165°F) and the edges are lightly browned. The chicken should be tender and the spices should have coated every piece. Once done, transfer to a plate or bowl and taste for seasoning, adjusting salt and pepper as needed.
While the chicken finishes cooking, prepare your fresh salad components—wash and tear the lettuce or greens into bite-sized pieces and set aside. If using broccoli, consider lightly steaming or blanching it briefly so it's tender but still crisp. Warm the tortillas in a dry skillet or over a gas flame for a few seconds per side—this makes them more pliable and easier to wrap without tearing.
Lay a warm tortilla flat on a clean surface. Create a line of ingredients down the center: start with a layer of hummus, then add some fresh salad greens, shredded cabbage from Step 1, a portion of the cooked chicken from Step 3, and any broccoli you're using. Drizzle with the yogurt sauce from Step 1. Fold the bottom edge of the tortilla up over the filling, fold in the sides, then roll tightly away from you like a burrito. The filling should be contained but not overstuffed—I've learned that a little restraint here prevents the wrap from falling apart when you bite into it.