Add more oil to the dutch oven and heat on medium. Add vegetables and a splash of chicken broth to the pot. Scrape up any brown bits on the bottom of the dutch oven, and cook the vegetables.
Whisk together the chicken broth and flour until no clumps remain and pour over the vegetables. Add the half n half as well, stir, and bring to a boil.
Nestle the chicken into the liquid and vegetables so the chicken breasts are mostly submerged in the liquid. Sprinkle with fresh thyme leaves, then cover and bake.