Brown the sausage in a large pot, breaking it apart as it cooks, until it reaches a browned color; move it to a plate lined with paper towels.
In the same pot, melt 1 tablespoon of the butter, cook the mirepoix for 5–7 minutes, then incorporate the garlic and cook for 30 seconds more.
Introduce the potatoes, parsley, basil, salt, and pepper; add the chicken stock, bring to a boil, then cover and simmer until the potatoes become tender.
In another pot, melt the remaining 4 tablespoons butter, whisk in the flour, then slowly pour in the milk, whisking until it thickens; blend in the heavy cream.
Merge the milk mixture into the soup, add the Cheddar and stir until it melts, fold in the sour cream, and put the sausage back into the pot.
Serve hot accompanied by hearty buttered bread, if desired.