Heat vegetable oil in a skillet over medium heat until shimmering, about 1 minute. Add chicken and 1 tablespoon honey. Cook and stir until chicken is no longer pink in the center and juices run clear, 3 to 5 minutes. Cover and remove from heat.
Place lemon juice, soy sauce, and 2 tablespoons honey in a small bowl; whisk dressing until well combined.
Rinse bok choy, broccolini, and snow peas together in a strainer. Transfer to a microwave-safe bowl; cook in the microwave until lightly steamed, about 3 minutes. Transfer to a large salad bowl; add tomatoes, mango, avocado, bell pepper, and chicken. Pour in dressing and toss well to coat. Serve with mint.