Combine chicken broth, 1½ tablespoons cornstarch, 2 tablespoons soy sauce, 2 tablespoons sake, 2 teaspoons honey, chili paste, and sesame oil in a bowl, whisking until smooth. Set sauce aside.
In a large bowl, whisk together 1½ teaspoons cornstarch, 1 tablespoon soy sauce, 1 tablespoon sake, and 2 teaspoons honey. Add chicken pieces and toss to coat evenly.
Heat 1½ tablespoons peanut oil in a wok or large pan over medium-high heat. Add marinated chicken and sauté until cooked through and golden.
Remove chicken from pan and wipe pan clean with paper towel.
Add remaining 1 tablespoon peanut oil to pan. Stir in green onions, ginger, and garlic; sauté for 30 seconds until fragrant.
Add bell pepper and carrot; sauté for 2 minutes until slightly softened.
Add snow peas and sauté for 1 minute until bright green and crisp-tender.
Return chicken to pan and stir in the prepared sauce mixture.
Bring to a boil and cook for 1 minute, stirring constantly, until sauce thickens and coats ingredients.
Serve immediately over hot cooked rice.