In a large pot or Dutch oven, melt the butter over medium heat. Add the diced onion, carrots, and celery. Cook, stirring occasionally, until softened, about 8-10 minutes.
Add the minced garlic and cook for 1 minute more until fragrant.
Whisk in the flour until a smooth paste forms. Cook for 1 minute, stirring constantly.
Gradually whisk in the vegetable broth until smooth. Bring the mixture to a simmer, stirring frequently, until it begins to thicken.
Stir in the milk (or half-and-half), cooked wild rice, dried thyme, black pepper, and salt. Cook for another 5-10 minutes, or until the soup is heated through and has reached your desired consistency.