Bring a medium pot of water to a rolling boil. Add the 6 oz pad thai rice noodles and cook according to package directions (about 3–6 minutes). Drain the noodles and rinse with cold water to stop cooking; set aside.
In a large pot, heat 1 Tbsp of the 2 Tbsp oil (olive or canola) over medium-high heat.
Add 1 cup finely chopped yellow onion and sauté until softened, about 3 minutes.
Add 1/2 Tbsp oil, 1 1/2 Tbsp minced garlic (about 4 cloves), 2 tsp curry powder, 2 tsp red curry paste, 1/2 tsp turmeric, and 1/2 tsp ground coriander. Cook, stirring, until fragrant, about 1 minute.
Add the remaining 1/2 Tbsp oil and 1 red bell pepper (sliced into small, thin strips). Sauté about 1 minute more.
Pour in 6 cups low-sodium chicken broth and bring the mixture to a boil.
Stir in 1 (13.5 oz) can coconut milk. Cover, reduce heat to low, and simmer gently for 10 minutes.
Stir in 3 cups shredded cooked rotisserie chicken, 8 oz snow peas (trimmed and cut in half crosswise), 2 Tbsp packed light-brown sugar, 2 Tbsp fish sauce, and 1/2 cup chopped green onions. Taste and season with salt as needed. Cook for 2 minutes to heat through.
Add 5 oz fresh spinach (about 4 packed cups), 1/2 cup chopped fresh cilantro, and 2 Tbsp fresh lime juice. Stir until the spinach wilts, about 1–2 minutes.
Add the cooked rice noodles to the pot and toss gently to combine and heat them through.
Serve the soup warm, topped with chopped peanuts and the 1 red chili pepper (seeded and chopped) or sriracha, to taste.