Begin by rinsing the wild rice under cold water. This helps remove any dust or debris.
In a large pot or Dutch oven, heat the olive oil over medium heat. Add the diced onion and sauté for about 5 minutes until it becomes translucent.
Add the minced garlic, diced carrots, and diced celery to the pot. Cook for another 5 minutes, stirring occasionally.
Stir in the diced potato and sliced mushrooms. Cook for an additional 5 minutes until the mushrooms soften.
Add the rinsed wild rice, vegetable broth, dried thyme, rosemary, bay leaf, salt, and pepper. Bring to a boil.
Once boiling, reduce the heat to low, cover the pot, and let it simmer for about 30 minutes, or until the wild rice is tender.
After 30 minutes, stir in the chopped kale or spinach, heavy cream or coconut milk, and lemon juice. Cook for an additional 5 minutes until the greens have wilted.
Remove the bay leaf, taste the soup, and adjust the seasoning with more salt and pepper if needed.
Serve hot in bowls, optionally garnished with fresh herbs or a sprinkle of black pepper.