Bring a medium pot of water to a boil. Generously season with salt. Add the noodles and stir to prevent sticking. Remove from heat and let soak until tender, 13 to 15 minutes. Drain in a colander and rinse with cold water.

Meanwhile, in a medium bowl, whisk together peanut butter, chili sauce, tamari, sriracha, lime zest, lime juice, 1/4 teaspoon salt, and 1/4 cup warm water until smooth; set aside.

In a large skillet over medium-high heat, heat the oil. Add the chicken, season with salt, and cook, stirring often, until lightly golden in spots and almost cooked through, about 4 minutes. Add the bell pepper, garlic, white parts of scallions, and ginger. Season with salt, if needed. Reduce heat to medium and cook, stirring often, until the peppers are tender, about 3 minutes more.

Add the noodles to the pan and toss well to combine. Remove from heat and add the reserved sauce. Toss to coat in the sauce.

Divide the noodles among bowls. Top with green parts of the scallions, cilantro, basil, and peanuts. Serve with lime wedges alongside.