In a large dutch oven or heavy-bottomed pot, melt the butter over medium heat. Add the diced onion, carrots, and celery. Cook for 5 to 7 minutes until the vegetables have softened and the onion becomes translucent.
Add the sliced mushrooms and minced garlic to the pot. Cook for another 8 to 10 minutes, stirring occasionally. The mushrooms will first release their moisture and then begin to brown, which is essential for a deep, savory flavor.
Stir in the dried thyme, rosemary, salt, pepper, and the uncooked wild rice. Toss everything together for a minute to toast the rice slightly, then pour in the broth and add the bay leaf.
Bring the mixture to a boil, then reduce the heat to low. Cover the pot with a lid and simmer for 45 to 50 minutes. The soup is ready when the wild rice is tender and some of the grains have started to “pop” open.
In a small bowl, whisk together the heavy cream and flour until smooth. Slowly pour this mixture into the simmering soup while stirring constantly. Let it cook for another 3 to 5 minutes until the soup thickens slightly and becomes beautifully creamy.
Remove the bay leaf. Taste the soup and adjust the salt and pepper if needed. Garnish with fresh chopped parsley before serving hot.