Slice chicken breasts into thin strips about half an inch thick. In a medium bowl, whisk together sweet chili sauce, soy sauce, lime juice, minced garlic, and grated ginger. Add chicken strips and toss to coat thoroughly. Let marinate for at least 15 minutes at room temperature, or up to 4 hours refrigerated.
While chicken marinates, prepare the creamy spread. In a small bowl, combine mayonnaise, 2 tablespoons sweet chili sauce, and 1 tablespoon lime juice. Whisk until smooth and set aside.
Heat 1 tablespoon oil in a large skillet or wok over medium-high heat until shimmering. Add chicken strips, reserving excess marinade in the bowl. Cook chicken for 5-7 minutes, stirring occasionally, until cooked through and developing caramelized edges.
Pour reserved marinade into the skillet with cooked chicken. Bring to a simmer and cook for 1-2 minutes, stirring constantly, until sauce thickens and coats chicken in a glossy glaze. Remove from heat and let cool slightly.
Warm tortillas for 20-30 seconds in microwave between damp paper towels, or heat one at a time in a dry skillet over medium heat for about 10 seconds per side until soft and pliable.
Lay a warm tortilla flat on clean surface. Spread 1 tablespoon of creamy chili sauce down the center, leaving a 2-inch border at top and bottom. Layer with shredded lettuce, red cabbage, julienned carrot, and cucumber matchsticks.
Place a generous portion of glazed chicken down the center on top of vegetables. Sprinkle with fresh cilantro and sliced green onions.
To fold the wrap, bring left and right sides of tortilla inward over the filling. Starting from the bottom edge closest to you, tightly roll the tortilla upward while tucking filling in as you go, creating a secure burrito-style wrap.
Slice each wrap in half on a diagonal. Serve immediately while chicken is still warm.
