For each wrap, place a tortilla on a flat work surface and cover half the tortilla surface with a piece of leaf lettuce. Top lettuce with 1 mozzarella slice, and distribute 1/3 cup shredded chicken over cheese.
Cut cucumber in half crosswise. Cut one half into 6 to 8 cucumber spears, and set aside. Use a vegetable peeler to shave remaining half of cucumber into very thin pieces, avoiding seeds. Place 1/2 of shaved cucumber on top of chicken.
Add half the avocado slices and half the sweet onion rings. Drizzle with 1 tablespoon green goddess dressing. Begin rolling the tortilla from the filled edge to the unfilled edge. Roll wrap in a paper towel to mold. Meanwhile, repeat steps with remaining ingredients for second wrap.
Release first wrap from paper towel and tie closed with a green onion leaf or chive. Release and tie second wrap.
To serve, place wraps on a bed of leaf lettuce. Lettuce leaves may be used for an optional outer wrap. Garnish with reserved cucumber spears and grape tomatoes.
