Preparation of Wild Rice: In a medium stockpot, bring 3 cups of chicken broth to a vigorous boil. Introduce the wild rice, then lower the heat to a moderate simmer. Securely place the lid on the pot and allow the rice to cook until it reaches an al dente texture, typically around 45-50 minutes, occasionally checking to ensure sufficient broth levels and to prevent the bottom from scorching. If necessary, add a small amount of broth. Once cooked, fluff the rice with a fork and set aside.
Sautéing Vegetables: In a larger stockpot, combine olive oil and butter and heat over medium settings. Add the chopped onions, carrots, celery, and sliced mushrooms to the pot. Season with onion powder, rosemary, thyme, kosher salt, and black pepper. Sauté the mixture for approximately 5-6 minutes, or until the vegetables begin to soften. Incorporate the minced garlic in the final 30 seconds to avoid burning.
Creating the Roux: Sprinkle the all-purpose flour over the sautéed vegetables, stirring continuously for about 3 minutes to fully integrate the flour without clumping.
Thickening with Broth: Gradually add around 8 cups of chicken broth to the vegetable and flour mixture, increasing the heat to bring the mixture to a boil. Once boiling, reduce the heat to maintain a gentle simmer, stirring consistently until the mixture thickens, which should take about 15-20 minutes.
Final Assembly: Add the previously cooked wild rice to the thickened mixture, followed by the milk and half and half. Continue to simmer the entire concoction for an additional 10-15 minutes, allowing the flavors to meld.
Serving: Serve the soup promptly while hot, ensuring it is thoroughly heated and well-combined.