Prep the vegetables: Dice the onion, slice the carrots and celery, and mince the garlic. If using mushrooms, slice them.
This can be done a day ahead and stored in the fridge.
Optional sauté for extra flavor: In a skillet, warm olive oil over medium heat. Sauté onion, carrots, celery, and mushrooms with a pinch of salt for 5–7 minutes until softened. This step adds depth but isn’t required.
Load the slow cooker: Add chicken, wild rice, vegetables, garlic, thyme, parsley, bay leaf, salt, pepper, and chicken broth to the slow cooker.
Stir to combine and make sure rice is submerged.
Cook low and slow: Cover and cook on Low for 6–7 hours or on High for 3–4 hours, until the rice is tender and the chicken shreds easily. Wild rice can vary, so check for doneness.
Shred the chicken: Remove the chicken to a plate and shred with two forks. Discard the bay leaf.
Make the creamy base: In a small saucepan, melt butter over medium heat.
Whisk in flour and cook for 1 minute, stirring constantly. Slowly whisk in milk (or half-and-half) until smooth and slightly thickened, 2–3 minutes.
Combine and finish: Stir the shredded chicken and the creamy mixture back into the slow cooker. If using greens, add them now and cook for 5–10 minutes until wilted.
Adjust seasoning with salt and pepper. Add a squeeze of lemon for brightness if you like.
Serve: Ladle into bowls. It’s great with crusty bread, crackers, or a simple side salad.