Heat the olive oil in a large pot or Dutch oven over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Add the minced garlic, ground ginger, ground cinnamon, salt, and black pepper. Stir and cook for another 1-2 minutes until fragrant.
Add the rinsed wild rice to the pot and stir to coat the rice with the spices and oil.
Pour in the vegetable broth and bring the mixture to a boil. Once boiling, reduce heat to low, cover, and simmer for about 25 minutes or until the rice is tender and cooked through.
Stir in the pumpkin puree and coconut milk, mixing well. Let simmer for an additional 5 minutes to combine the flavors.
Add the chopped kale or spinach to the pot and cook until wilted, about 2-3 minutes. If desired, stir in maple syrup for a touch of sweetness.
For a creamier texture, you can blend a portion (or all) of the soup using an immersion blender or transferring it to a regular blender in batches.
Serve hot in bowls, garnished with additional kale or a drizzle of coconut milk if desired.