Cook the vegetables. Melt 1 1/2 tablespoons butter in a large soup pot over medium heat. Add the carrots, celery, and onion. Cook for about 5 minutes, stirring occasionally, until they begin to soften.
Add the mushrooms. Stir in the sliced cremini mushrooms and cook for another 5 to 7 minutes. Let them release their moisture and continue cooking until they begin to brown.
Add the chicken, rice, and broth. Pour in the chicken stock, then add the chicken, brown rice and wild rice blend, rosemary, thyme, celery seed, and bay leaves. Season lightly with salt and black pepper and stir well.
Simmer. Bring the soup to a gentle simmer, then cover the pot. Cook according to the rice blend’s package cooking time, beginning to check it around 30 minutes. The rice should be tender with a little chew, and the chicken should reach 165°F in the center. If the rice absorbs too much liquid before it is tender, add a little more chicken stock.
Add the milk and cream. Remove the bay leaves and reduce the heat to low. Stir in the whole milk and whipping cream. Let the soup warm gently without bringing it to a hard boil.
Prepare the roux. Melt 3 tablespoons butter in a small saucepan over medium-low heat. Add the flour and cook for 1 to 2 minutes, stirring constantly. Gradually whisk in about 1 cup of hot broth from the soup until smooth.
Thicken the soup. Slowly stir the roux mixture back into the soup. Cook gently for 3 to 5 minutes, stirring occasionally, until the broth is creamy and lightly thickened.
Finish with lemon. Remove the pot from the heat. Stir in the lemon zest and fresh lemon juice. Taste and add more salt or black pepper if needed. Let the soup sit for about 5 minutes before serving.
