Heat the olive oil in a large Dutch oven over medium-high heat. Add the diced onion, carrots, celery, mushrooms, and thyme. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring occasionally for 8-10 minutes, until the vegetables soften and the mushrooms release their moisture.

Add the wild rice and toast for 1-2 minutes, stirring occasionally, until it releases a nutty aroma. Pour in the chicken broth and add the chicken breasts. Bring the mixture to a boil. Season with the remaining salt and pepper. Reduce the heat to a simmer, cover the pot, and cook for 30 minutes.

Remove the chicken from the pot and set it on a cutting board. Shred the chicken using two forks.

Continue cooking the soup for an additional 15 minutes until the rice is tender. Return the shredded chicken to the pot and stir to combine.

In a blender, combine the white beans and 1 cup of water. Process until smooth. Add the blended bean mixture to the soup, stir, and cook for another 5 minutes. Taste and adjust seasoning if necessary before serving.