Heat the oil in a large saucepan or Dutch oven or medium-high heat. Stir in the shallot and sweet potatoes. Cook 5 minutes, stirring occasionally, until the shallots are softened.
Add the garlic, turmeric and cayenne and stir until fragrant, about 30 seconds. Add the peanut butter, soy sauce, broth and ½ cup of the coconut milk. Bring to a simmer, then lower the heat and partially cover the pan with a lid. Cook 20 minutes, or until the sweet potatoes are very soft.
Scoop out 1 cup of the soup and puree in a blender or use an immersion blender right in the pot. Stir in the remaining ½ cup coconut milk, kale and peanut sauce.
Cook the noodles according to package directions. Drain and divide into serving bowls.
Ladle the soup over the noodles, sprinkle with peanuts and serve.