Add 2 tablespoons butter to a Dutch oven and place over MEDIUM-HIGH heat. Add the mushrooms and cook, stirring, occasionally, until they are tender and browned and have absorbed most of the liquid they've let off, about 5 to 6 minutes. Add Worcestershire and cook until it is absorbed by the mushrooms, 1 to 2 minutes more. Remove from heat and transfer the mushrooms to a plate and set aside.
Melt the remaining butter in the same Dutch oven over MEDIUM-HIGH HEAT. Add the onion, celery, and carrot and cook, stirring occasionally for about 5 minutes, or until veggies have softened. Add the garlic, thyme leaves, rubbed sage, smoked paprika, salt and pepper, and cook for another minute or two, stirring the mixture well to coat the veggies with the spices.
Reduce the heat under the Dutch oven to MEDIUM and sprinkle the flour over the vegetables. Cook, stirring for 2 to 3 minutes. Slowly pour in the chicken broth, stirring as you add it to ensure the mixture is smooth. Add the uncooked rice and return the mushrooms to the pot. Increase the heat as needed to bring the soup to a boil, stirring occasionally. Cover, turn the heat down to LOW and simmer for 45 to 50 minutes or until the rice is tender.
Add the half and half or whole milk. Stir until it is well incorporated and remove from the heat. Taste and season with additional salt and pepper, if needed. Garnish with additional thyme leaves and serve.