Bring a medium pot of water to a boil; generously season with salt. Add 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.

In a medium bowl, whisk peanut butter, chili sauce, tamari, sriracha, lime zest, lime juice, ¼ teaspoon salt, and ¼ cup warm water until smooth; set aside.

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

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

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