Heat the olive oil in a large soup pot over medium heat. Remove the sausages from their casings, add the sausage meat to the pot, and break it up with a spatula. Fry until the sausage is golden on the outside and cooked through, then remove to a separate plate. If the sausages are very fatty, pour off any excess fat, leaving about 2 tablespoons in the pot for flavor.

To the same pot, add the chopped onion, sliced celery, and sliced carrots. Sauté over low heat for about 10 minutes until softened. Stir in the minced garlic and dried thyme; cook for 30 seconds until fragrant. Sprinkle the flour evenly over the vegetables and fat, stirring to make a paste (roux). Cook the roux for a couple more minutes, then return the cooked sausage from Step 1 back into the pot.

Pour a little chicken or vegetable broth into the pot, stirring well to combine with the roux and avoid lumps. Gradually add the remaining broth, mixing well. Drop in the bay leaves and wild rice. Bring the soup to a simmer over low to medium heat and cook for 30 minutes, allowing the flavors to meld and the rice to begin softening.

Add the peeled and chunked potatoes to the simmering soup. Continue to cook for about 15 minutes longer, or until both the wild rice and potatoes are tender and fully cooked.

Pour in the light cream or half and half, stirring until well combined. Season the soup with salt to taste and continue to heat gently until the cream is warmed through, but don't let it boil. Take off the heat and serve hot. I like to add a sprinkle of chopped fresh parsley on top for a touch of color, though it's optional!