Heat the olive oil in a large heavy pot over medium heat. Add the mushrooms and cook, stirring occasionally, until they release their liquid and brown at the edges, 6β8 minutes.
Add the onion, celery, and carrot and cook until softened, 6β8 minutes. Stir in the garlic and cook until fragrant, 30β60 seconds.
Stir in the wild rice and cook to toast lightly, about 2 minutes.
Add the vegetable broth, dried thyme, and bay leaves. Bring to a boil, then reduce to a gentle simmer, partially cover, and cook, stirring occasionally, until the rice is tender and many grains have split, 45β55 minutes; discard the bay leaves at the end of simmering.
Meanwhile, make the roux: In a medium saucepan over medium heat, melt the unsalted butter. Whisk in the flour and cook, whisking constantly, until blond and nutty, 2β3 minutes. Gradually whisk in the whole milk until smooth and thickened, 3β5 minutes; remove from the heat.
Whisk the milk-roux mixture into the soup. Simmer gently, stirring, until slightly thickened and creamy, 5β10 minutes.
Stir in the heavy cream and season with the kosher salt and black pepper. If the soup is too thick, loosen with a splash of hot water to desired consistency.
Ladle into bowls and garnish with the flat-leaf parsley. Serve hot.