Add 3 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 released most of their liquid, 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 vegetables have softened.
Add the garlic, thyme leaves, ground sage, smoked paprika, salt, and pepper, and cook for another two minutes, stirring well to coat the vegetables with the spices.
Reduce the heat to medium-low heat and sprinkle the flour over the vegetables. Cook and continue to stir 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 and bring 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 whole milk. Stir until it is well incorporated and remove from the heat.
Taste and season with additional salt and pepper, if needed.