Bring a large pot with lightly salted water to a rolling boil. Drop the udon in a few noodles at a time and return to a boil. Cook uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, 10 to 12 minutes. Drain; return to the pot.
While the udon noodles are cooking, whisk the chicken broth, ginger, soy sauce, peanut butter, honey, chile oil, and garlic in a saucepan over medium-high heat. Bring to a boil, whisking until the peanut butter has melted. Pour the sauce over the noodles. Add the chicken and red bell pepper; toss until the noodles are evenly coated in the sauce. Sprinkle with green onions, chopped peanuts, and cilantro to serve.