For the Sauce:Place soy sauce, peanut butter, 2 tablespoons sesame seeds (reserve remaining sesame seeds for later), garlic, vinegar, brown sugar, ginger, and Sriracha in the bowl of a food processor or blender and process for 30 seconds, or until smooth. With unit running, gradually pour in the hot water. Continue to process for a few more seconds. Transfer sauce to a container, cover, and refrigerate.
For the Chicken:Heat vegetable oil in a skillet over medium-high heat. Add chicken and season to taste with salt and fresh ground pepper. Cook chicken for about 5 minutes until nicely browned on one side. Flip the chicken and carefully add 1/2 cup water to the skillet, cover, and reduce heat to medium-low. Continue to cook for another 6 to 8 minutes, or until thoroughly cooked through to the center. Transfer the chicken to a cutting board and allow to cool. Once cool enough to handle, use forks to shred into bite size pieces.
For the Noodles:While chicken is cooking, bring water to boil in a big pasta pot. Add noodles and cook for length of time indicated on package, stirring occasionally. Drain noodles and then rinse with cold water and drain again, leaving them just slightly wet. Transfer to a large bowl and toss with sesame oil.
Add shredded chicken, green onions, carrot, and bell pepper to noodles and toss to combine.
To Finish and Serve:Add dressing to noodles and toss until combined. Season with salt and fresh ground pepper, to taste. Sprinkle with cilantro and remaining sesame seeds and serve.
Make-Ahead Instructions:Refrigerate dressing and noodle/veggie mixture separately for up to two days. When ready to serve, remove dressing and noodles from refrigerator. If dressing has thickened to much, stir in about 1 tablespoon of hot water to thin it slightly. Add dressing to noodles and proceed as directed in recipe above.