Drizzle the oil into a 5 QT Dutch oven or large soup pot and bring to medium heat. Add the carrots, celery, onions and garlic and stir to coat in the oil. Cook, stirring occasionally, for 5-7 minutes until the onions are softened.
Add the broth, chicken breasts, potatoes, bouillon, poultry seasoning, oregano, rosemary, thyme, and the bay leaf to the pot. Stir together and make sure the chicken is covered by broth. Bring the pot to a boil and then reduce the heat to low to simmer. Partially cover the pot with the lid on a bit offset/crooked so that there’s about a 1” opening on the side. Simmer for 25 minutes.
When there’s about 10 minutes left on the simmering chicken, stir together the flour, pepper, and salt in a small dish. Place the butter in a medium saucepan and melt it over medium heat. Slowly whisk the flour mixture into the melted butter until combined and cook the resulting paste for 1-2 minutes. Slowly whisk the milk in a little bit at a time until smoothly combined with the flour/butter mixture. Once smooth, cook, whisking a little more than occasionally, until thickened and lightly bubbly. Whisk again and remove from heat.
When the timer goes off for the big pot, uncover and remove the chicken breasts to a cutting board. Leave the rest of the pot cooking uncovered on low. Shred the chicken on the cutting board using two forks.
Locate and remove the bay leaf from the pot. Stir the creamy mixture you made in step 3 into the main pot until well combined. Add the chicken back into the pot as well and stir. Cook for another 5-10 minutes, stirring occasionally.