Heat one tablespoon olive oil over medium heat in a large pot. Add chicken; season with salt and pepper. Cook and stir until cooked through, about 3 minutes. Transfer to a bowl.
Heat remaining olive oil in same pot over medium heat. Add onion, lightly sprinkle with salt to sweat liquid and keep from browning; cook and stir 2 to 3 minutes until softened. Add carrots, celery, and potato; cook and stir 2 minutes more. Add thyme and chicken broth and bring to boil. Reduce heat and keep at a high simmer 10 minutes or until vegetables are nearly tender.
Meanwhile, heat a small saucepan over medium heat. Add butter. Once butter melts, add flour all at once. Whisk constantly for a minute or two, just to cook out the raw flour taste. Add mixture to vegetables in broth and stir to combine. Boil until mixture starts to thicken, about 3 minutes. (At this point, the potatoes should be tender.)
Stir in cooked chicken and half and half; season to taste with salt and freshly ground black pepper.
Serve hot as is or with Pie Crust Crackers if desired.
If you like peas in your pot pie, I recommend tossing in one cup of frozen peas near the end of step 2 when the other vegetables are almost tender.