Rinse the wild rice in a fine mesh strainer under cold water for 30-60 seconds, then set aside to drain.
Finely dice the onion, celery, and carrots to uniform 1/4-inch pieces. Mince the garlic and slice the mushrooms thinly.
In a large Dutch oven over medium heat, melt the butter. Add the onion, celery, and carrots and cook for 7-9 minutes until softened and fragrant.
Push vegetables to the sides, add garlic to the center, and cook for 30-60 seconds until fragrant. Add mushrooms and cook for 4-5 minutes until their moisture has evaporated.
Sprinkle dried thyme, rosemary, garlic powder, and pepper over the vegetables and cook for 30-60 seconds to bloom the herbs.
Add the flour and stir constantly for 1-2 minutes until the raw flour smell disappears and the mixture smells nutty.
Gradually pour in the broth while whisking continuously to prevent lumps. Add the rinsed wild rice and stir to combine.
Bring to a gentle boil, then reduce heat to medium-low, cover with the lid slightly ajar, and simmer for 40-50 minutes until the rice is tender and the grains have split open.
Reduce heat to low and stir in the heavy cream. Warm through for 3-5 minutes without letting the soup boil.
Taste and adjust seasoning with salt and pepper. Optionally blend 1-2 cups of soup with an immersion blender for a creamier texture. Serve hot, garnished with fresh thyme or parsley.