In a medium bowl, add soy sauce, chicken broth, honey, rice vinegar, peanut butter, sesame oil, garlic and cornstarch. Whisk and set aside.
In a large pot with boiling water, cook soba noodles until al dente, for about 2 - 3 minutes. Reserve about a cup of noodles' water (starchy water is great for diluting the sauce), transfer noodles to a colander, rinse with cold water and leave to drain.
Preheat non-stick wok or large deep skillet on high heat, add 1 tablespoon of oil to coat and add chicken. Cook for about 7-8 minutes until golden brown, stirring occasionally. Add 1/3 of prepared sauce and stir constantly for a minute or until sauce thickens and coats the meat. Transfer a bowl and set aside.
Add remaining 1 tablespoon of oil and bell peppers, cook until fragrant and some peppers are golden brown, about 3-4 minutes, stirring constantly.
Reduce heat to medium and add kale, previously cooked soba noodles and chicken, and remaining sauce. Toss gently until mixed, about 1 minute, and remove from heat. Sauce will thicken as it warms up. If noodles absorb too much sauce, add some reserved noodles' water.
Add peanuts, green onions and red peppers flakes. Serve hot.
Notes
Store: Refrigerate in an airtight container for up to 2 days. Do not freeze.
Reheat: Reheat in a microwave for 1 – 1.5 minutes or sti fry quickly on the stove on high heat with a bit of oil or chicken broth.