Heat the olive oil in your stockpot over medium heat, then add the sausage and break it apart with a wooden spoon. Cook until it's no longer pink and has a bit of golden color on the edges, about 6 to 8 minutes. Once browned, transfer the sausage to a plate and set it aside for later.
In the same pot, add the diced onion and cook until it turns soft and translucent, roughly 4 minutes. Stir in the minced garlic and thyme, cooking for another minute until fragrant. Don't rush this step, since it's what builds the soup's flavor foundation.
Pour in the chicken broth and add the cubed sweet potatoes, stirring everything together. Bring the pot to a gentle boil, then reduce the heat and let it simmer uncovered for about 15 minutes, until the potatoes turn fork-tender.
Return the browned sausage back into the pot along with the torn kale leaves. Let everything simmer together for another 5 minutes, just until the kale wilts down and turns a deep green color.
Lower the heat and stir in the heavy cream or coconut milk, letting it warm through without boiling. Season with salt and pepper to taste, give it one final stir, then ladle it into bowls while it's still hot.