Cook the noodles according to package directions until just al dente. Reserve 1/2 cup of the starchy cooking water, then drain and toss with a splash of oil to prevent sticking.
Make the peanut sauce: whisk peanut butter, soy sauce, curry paste or sriracha, brown sugar, rice vinegar, lime juice, and sesame oil.
Add warm water a little at a time until smooth and pourable, like heavy cream.
Season the chicken lightly with salt and pepper. Heat 1 tbsp oil in a large skillet or wok over medium-high heat. Cook chicken in a single layer until golden and cooked through, 4–6 minutes.
Transfer to a plate.
Add remaining oil. Sauté garlic, ginger, and the white parts of the green onions for 30 seconds until fragrant. Add bell pepper and carrots; stir-fry 2–3 minutes until crisp-tender.
Return chicken to the pan.
Add spinach and toss until just wilted.
Pour in the peanut sauce. Add a splash of reserved noodle water and stir until everything is glossy and well coated.
Add noodles and toss gently over low heat for 1–2 minutes. Adjust thickness with more noodle water as needed.
Taste and add more lime, soy, or heat to balance.
Serve hot, topped with peanuts, cilantro, green onion tops, sesame seeds, and lime wedges.