In a mixing bowl, whisk together lime juice, sesame oil, soy sauce, honey, minced garlic, and grated ginger. Set it aside.

Using a julienne peeler or spiralizer, transform carrots into noodle-like strands and massage with two-thirds of the dressing. Marinate for 5-10 minutes.

Thinly slice red bell pepper and julienne cucumber. Add to the large bowl with marinated carrots and toss to combine.

Divide the dressed vegetable mixture among serving bowls, top each with cooked chicken breast, and drizzle remaining dressing over.

Garnish with chopped cilantro or green onions and sesame seeds before serving.