Melt 1 tablespoon butter in a medium dutch oven over medium-high heat. Add the mushrooms and cook, stirring once or twice, until well browned, about 3 minutes. Reduce the heat to medium. Add the remaining 2 tablespoons butter and cook until melted. Add the leeks and carrots and season with 1 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the leeks are softened, about 3 minutes.
Add the thyme and stir to combine. Add the chicken broth, lemon zest, bay leaves and 3 cups water. Bring to a boil, then reduce the heat to medium to maintain a simmer. Add the turkey and wild rice; simmer until the rice is just tender, 13 to 15 minutes.
Add the spinach and simmer until wilted, about 1 minute. Discard the bay leaves and lemon zest and stir in the lemon juice; season with salt and pepper. Divide the soup among bowls.