Place the sweet potatoes into a large pot and cover with cold water. Bring to a boil; reduce heat to medium-low, cover, and simmer until tender, 6 to 8 minutes.
Drain and allow to steam dry for a minute or two. Puree potatoes in a food processor or blender in batches until smooth. Return pureed potatoes to the large pot; set aside.
Heat the oil in a large skillet over medium heat; cook the onion in the hot oil until translucent, about 4 minutes.
Add the sausage to the skillet and cook until completely browned, breaking the meat apart with the back of a wooden spoon as it cooks. Stir the garlic, mushrooms, and Italian seasoning into the sausage-and-onion mixture; continue cooking until the mushrooms are tender, another 4 to 5 minutes. Season with salt and pepper; transfer to the pot with the potatoes.
Place the large pot over low heat.
Pour the half-and-half over the mixture, stirring to mix. Allow the soup to slowly heat until hot and to your desired consistency.
