Coat the bottom of a large nonstick skillet with 2 tablespoons of olive oil.

Give the oil a minute to heat up, and then add some of the mushrooms to the skillet. Add only as many as you can fit without crowding, arranging them in an even layer.

Let the mushrooms cook for about 5 minutes, flip them, and cook them for about 5 minutes more, until tender and browned on both sides.

Remove the mushrooms from the skillet and transfer them to a plate.

Cook the remaining mushrooms in batches in the same manner, adding oil to the skillet as needed between batches, using up to 4 tablespoons total.

Coat the bottom of a large pot with the remaining tablespoon of oil and place it over medium heat.

Give the oil a minute to heat up, then add the onion.

Sweat the onion for about 5 minutes, until it becomes soft and translucent.

Add the garlic and continue cooking and stirring for about 1 minute, until very fragrant.

Stir in the wine, raise the heat, and bring it to a boil. Lower the heat and allow the wine to simmer until most of the liquid has cooked off, about 5 minutes.

Begin adding the flour, a bit at a time. Stir the flour in between each addition to form a paste that coats the onions and garlic. Cook everything for about a minute, stirring constantly.

Begin whisking in the broth, a bit at a time, until all of the broth has been added and the flour is blended in.

Stir in the coconut milk, thyme, and mushrooms.

Raise the heat and bring the liquid to a boil. Lower the heat and allow the soup to simmer for about 15 minutes, until the liquid thickens up a bit.

Remove the pot from the heat.

Transfer about half of the soup to a food processor or blender and blend it until smooth. Return it to the pot.

Reheat the soup if needed, then season it with salt and pepper to taste.

Ladle into bowls and serve.