Load the crockpot: Add wild rice, mushrooms, onion, carrots, celery, garlic, thyme, bay leaf, broth, soy sauce or tamari, salt, and pepper to the slow cooker. Stir to combine.
Cook low and slow: Cover and cook on Low for 6–7 hours or High for 3.5–4 hours, until the rice is tender and the vegetables are soft.
Make a quick roux: About 20 minutes before serving, melt butter (or warm olive oil) in a small saucepan over medium heat.
Whisk in flour and cook 2–3 minutes, stirring, until it smells toasty. Slowly whisk in the milk or half-and-half until smooth and slightly thickened, 3–4 minutes.
Finish the soup: Remove the bay leaf from the slow cooker. Stir in the creamy roux mixture and the lemon juice.
Taste and adjust seasoning with more salt and pepper as needed.
Serve: Ladle into bowls and top with chopped parsley. Add extra lemon if you like a brighter finish.