Heat olive oil in a skillet over medium heat. Sauté onion, garlic, carrots, and celery for 3–4 minutes until fragrant and slightly softened.
Transfer veggies to slow cooker. Add chicken breasts, diced potatoes, chicken broth, thyme, parsley, salt, and pepper. Stir to combine.
Cover and cook on low for 6–7 hours or high for 3–4 hours, until chicken is tender.
Remove chicken, shred with two forks, and return to pot. Stir in milk and Greek yogurt.
Whisk flour with a few tablespoons of broth, stir into slow cooker. Cook uncovered for 20–30 minutes until slightly thickened.
Add peas and corn (if using) during the last 15 minutes of cooking.
Taste and adjust seasoning. Garnish with fresh parsley before serving.
Notes
Use chicken thighs for more flavor. Add a splash of lemon juice at the end for brightness. Serve over mashed potatoes or in puff pastry shells for a twist.