In a large Dutch oven or heavy-bottomed pot, melt the butter over medium-high heat. Add the sliced mushrooms and cook without stirring much for the first 5 minutes to allow them to brown. Continue cooking until deeply golden brown and moisture has evaporated (about 10-12 minutes total).

Reduce heat to medium. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and fresh thyme, cooking for 1 minute until fragrant.

Sprinkle the flour over the vegetables and stir constantly for 1 minute to cook out the raw flour taste, creating a roux.

Slowly whisk in the broth, scraping up any browned bits from the bottom of the pot. Bring the mixture to a simmer, then add the diced potatoes. Season with salt and pepper. Reduce heat to low, cover partially, and simmer for 15-20 minutes, or until the potatoes are fork-tender.

Remove about 2 cups of the soup mixture (primarily mushrooms and broth, leaving most of the potatoes behind) and transfer to a blender. Blend until very smooth. Alternatively, use an immersion blender directly in the pot until the desired chunkiness is achieved, ensuring some potatoes remain intact.

Return the pureed mixture to the pot (if using a separate blender). Stir in the heavy cream. Heat gently until warmed through (do not boil after adding cream). Taste and adjust seasoning as needed.

Ladle the soup into bowls. Garnish generously with fresh chopped parsley before serving hot.