Melt butter in a large pot over medium heat; sauté onion until soft, about 5 minutes.
Add garlic and mushrooms; cook until mushrooms release moisture and brown, about 8–10 minutes.
Stir in wild rice, thyme, and bay leaf; pour in broth and bring to a boil.
Reduce heat, cover, and simmer 35–40 minutes until rice is tender.
If using oat milk, whisk flour into oat milk to smooth, then stir in; if using half-and-half, add directly and heat gently.
Simmer 5 more minutes to thicken, remove bay leaf, season to taste, and garnish.