Heat olive oil in a large skillet over medium heat. Add minced garlic and ginger, cooking for about a minute.
Add ground chicken, crumbling it as it cooks until fully browned, about 5-7 minutes.
Mix in soy sauce, hoisin sauce, and sesame oil, followed by shredded carrots and cabbage. Cook for an additional 2-3 minutes.
Season with salt and pepper, then remove from heat and let cool.
Heat vegetable oil in a deep skillet or pot over medium-high heat.
Spoon about 1 tablespoon of the chicken mixture into the center of a wonton wrapper, fold to create a triangle, and seal the edges.
Fry wonton bites in hot oil until golden brown and crispy, about 2-3 minutes per side.
Transfer cooked wonton bites to a plate lined with paper towels to absorb excess oil.
Serve warm, garnished with fresh cilantro and lime wedges.