Combine the broth, 1 1/2 tablespoons cornstarch, 2 tablespoons soy sauce, 2 tablespoons sake, 2 teaspoons honey, chile paste, and sesame oil, stirring with a whisk; set aside.
Peel shrimp, leaving the tails intact. Starting at the tail end, butterfly each shrimp, cutting to, but not through, underside of shrimp; remove and discard vein. Combine 1 1/2 teaspoons cornstarch, 1 tablespoon soy sauce, 1 tablespoon sake, and 2 teaspoons honey in a large bowl, stirring with a whisk. Add shrimp; toss gently to coat.
Heat 1 1/2 tablespoons peanut oil in a large nonstick skillet or a wok over medium-high heat. Add shrimp mixture; sauté 4 minutes or until shrimp are done. Remove shrimp from pan. Wipe the pan clean with paper towels. Add 1 tablespoon peanut oil to pan. Stir in onions, ginger, and garlic, and sauté 30 seconds. Add bell pepper and carrot; sauté 2 minutes. Add snow peas; sauté 1 minute. Stir in shrimp and broth mixture. Bring to a boil; cook mixture 1 minute or until slightly thick, stirring constantly. Add noodles, and cook 1 minute or until thoroughly heated.