In a Dutch oven or heavy-bottomed medium pot over medium heat, add Italian sausage. If you are using bulk sausage without the casing you can use a potato masher to break up the sausage into pieces. Cook until browned and cooked through. Remove sausage pieces with a slotted spoon and transfer to a bowl. Set aside while you continue with the rest of the recipe.
Add oil to the pot if needed. Stir in the onions and fennel; cook until onions are tender and translucent, about 5 minutes. Add the garlic, paprika, and crushed red pepper and saute until fragrant, about 1 minute.
Add the potatoes, chicken stock, salt and pepper. Start with 1 1/2 teaspoons of salt if using unsalted stock. Bring to a boil over high heat, lower to a simmer and cover until potatoes are fork tender, about 15-20 minutes.
Uncover, increase the heat to medium and stir in the coconut milk (including any cream that has settled to the top) and the cooked sausage. Stir until milk has dissolved. Add the kale into the soup just before serving and cook until softened, 2-3 minutes. Season with salt and pepper to taste.