Heat oil in a large Dutch oven or soup pot over medium heat until it begins to shimmer. Add the onion and cook for 2 to 3 minutes, or until translucent and tender.
Add the garlic, thyme, parsley, and potatoes and cook an additional 30 seconds, stirring once or twice.
Quickly whisk in flour and cook another 30 seconds, stirring continuously. Pour in the half-and-half, chicken broth, cream, and bay leaf and bring to a boil.
Reduce heat to simmer. Stir in chicken and veggies and continue to simmer for 30 minutes, or up to 2 hours (check often to ensure liquid does not cook off).
Remove and discard bay leaf. Season to taste with salt and pepper (I like 1 teaspoon salt and 1/2 teaspoon pepper). Serve warm with buttermilk biscuits.
Notes
Cooked chicken and chicken broth: You can make both at the same time, completely from scratch with this recipe for homemade chicken broth.
Make it even thicker: Take 3 tablespoons of cornstarch and mix it with a small amount of cold water to make a slurry. Add it to the soup as it cooks. In a few minutes, it will be even thicker.
Cook it ahead of time: You can make this soup a few days before you need it and gently rewarm on the stove or on LOW in a slow cooker.
Freezing: Usually, cream-based soups don't hold up as well in the freezer, because the dairy gets a grainy texture as it thaws out and gets reheated. In other words, make this recipe to enjoy sooner rather than later.