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, bay leaf and salt and bring to a boil.
Reduce heat to a 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. Serve warm with buttermilk biscuits cut into squares and sprinkled on top.
Notes
Soup can be lightened up by replacing the cream with milk. Source: slightly adapted from the Oh Sweet Basil Kitchen cookbook with permission to publish. I reduced the thyme per the taste preferences of my family. The original recipe calls for 5 teaspoons of thyme.