In a large pot, heat half of the olive oil over medium-high heat. Add the sliced shiitake mushrooms and chopped poblano chile, sautéing them for about 5-6 minutes until they turn golden and fragrant. Remove them and set aside.

In the same pot, heat the remaining olive oil and add the turkey breast, skin-side down. Brown for about 5 minutes, then season with salt and pepper.

Add cold water, chopped onion, celery, carrot, and minced garlic to the pot. Stir to combine and bring to a gentle simmer. Cook for about 1 hour.

Carefully remove the turkey after an hour and add the quartered baby potatoes. Simmer for another 25-30 minutes until the potatoes are tender.

Once cool, shred the turkey and return it to the pot along with the sautéed mushroom mixture and fresh thyme. Stir thoroughly and adjust seasoning.

In a skillet, melt the butter and add the all-purpose flour. Whisk for 1-2 minutes until pale. Gradually whisk into the soup.

Slowly stir in the heavy cream and simmer for an additional 3 minutes.

Ladle the soup into bowls and garnish with fresh parsley or thyme. Serve alongside crusty bread or a light salad.