In a large pot or Dutch oven, heat the olive oil over medium heat. Add the chopped onion, carrots, and celery. Sauté for about 5 minutes, until the vegetables are softened.
Add the minced garlic and sauté for an additional minute until fragrant.
Remove the casing from the Italian sausage and crumble it into the pot. Cook, stirring occasionally, until browned, about 5-7 minutes.
Stir in the diced potatoes, chicken broth, dried thyme, salt, and pepper. Bring the mixture to a boil, then reduce the heat and let it simmer for 15-20 minutes, or until the potatoes are tender.
Slowly pour in the milk and stir to combine. Allow the soup to heat through for another 5 minutes without boiling.
Taste and adjust seasoning as needed. If desired, garnish with fresh parsley before serving.