In a large pot over medium-high heat, heat 3 tablespoons oil. Cook mushrooms and chile, stirring occasionally, until softened and browned, 5 to 6 minutes; season with salt and pepper. Transfer mushroom mixture to a medium bowl; set aside.

In same pot over medium-high heat, heat remaining 2 tablespoons oil. Cook turkey skin side down, turning occasionally, until golden brown, about 5 minutes; season with salt and pepper. Add water, onion, celery, carrot, and garlic. Bring to a simmer and cook 1 hour.

At the 1-hour mark, transfer turkey to a cutting board and let cool slightly. Add potatoes to pot and cook, stirring occasionally, until tender, 25 to 30 minutes.

Cut turkey meat off bone and shred into bite-size pieces. Add reserved mushroom mixture, turkey meat, and thyme to broth; season with salt and pepper.

In a medium skillet over medium heat, melt butter. Add flour and whisk to dissolve. Cook, whisking constantly, until blonde in color, 1 to 2 minutes. Let cool slightly.

Add roux to soup, whisking to avoid clumps. Pour in cream, bring to a low simmer, and cook, stirring occasionally, until heated through, about 3 minutes.

Divide soup among bowls. Serve with Worcestershire sauce alongside.