Melt butter over medium-high heat. Add mushrooms, season with salt and pepper, and cook, stirring frequently, until liquid has evaporated and mushrooms are well-browned, about 12 minutes total.
Add onion and cook, stirring, until softened, about 3 minutes.
Add garlic and cook until fragrant, about 30 seconds.
Add flour and stir to combine.
Add white wine and cook until reduced by about half, scraping up browned bits from the bottom of the pan.
Add milk, chicken stock or water, bay leaves, and thyme sprigs and stir to combine. Bring to a bare simmer and cook for 20 minutes.
Remove herbs. Blend soup. Season to taste with more salt and pepper and a squeeze of lemon juice (if desired). Serve immediately, garnished with minced herbs and olive oil.
