Combine mushrooms, 1 teaspoon oil, and ¼ teaspoon salt in skillet. Cover and cook over medium-low heat until soft, ~10 minutes. Uncover, increase heat to medium-high, cook until browned 10-12 minutes. Transfer to food processor and pulse until finely ground.

Heat remaining 1 teaspoon oil over medium heat. Add onion and mushrooms, cover, cook until onion softens, ~10-12 minutes. Stir in chili powder and cook ~30 seconds.

Add tomato sauce, ketchup, water, Worcestershire, sugar, and vinegar. Bring to simmer and keep heat low and reduce until sauce thickens, ~15 minutes.

Add ground turkey, break up meat, and simmer until cooked through, ~10 minutes. Season with salt and pepper and serve on buns.