In a large pot, heat the vegetable oil over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Stir in the minced garlic and grated ginger, cooking for an additional minute until fragrant.
Add the curry powder and turmeric, stirring well to combine and toast the spices for about 1 minute.
Pour in the coconut milk, vegetable broth, and milk. Stir in the soy sauce and brown sugar. Bring the mixture to a gentle simmer.
Add the udon noodles to the pot and cook according to package instructions, usually about 5-7 minutes, until tender.
Stir in the baby spinach and cook for another 2 minutes until wilted. Season with salt and pepper to taste.
Serve hot, garnished with chopped green onions.