Bring a pot of water to a boil over medium-high. Add rice, and reduce to a simmer. Cover and cook 40 to 50 minutes, until the rice has burst open and is tender. Drain, and set aside.
Heat oil in a Dutch oven or stock pot over medium heat. Add onions, carrots, celery, and 1/2 tsp. of the salt; cook 8 to 10 minutes, until softened. Add garlic, mushrooms, thyme sprigs, black pepper, and remaining 1 tsp. salt. Cook, stirring occasionally, until the mushrooms have released all their liquid and turn golden-brown, about 15 minutes.
Add flour and stir until the vegetables become sticky and there is no more visible dry flour. Increase heat to medium-high and add wine. Stir and scrape the brown bits from the bottom of the pan. Continue simmering until the wine has reduced by about half.
Add vegetable broth, bring mixture to a boil, then reduce to a simmer. Simmer, uncovered, for 20 minutes.
Add milk and cooked wild rice; stir to combine. Simmer for another 10 minutes. Remove leftover thyme sprigs. Garnish with cheese, if desired.