Heat the oven to 350F.
Heat a large skillet over medium low heat. Add 1 tablespoon butter and 1 tablespoon olive oil. Add the bread cubes, sprinkle with ¼ teaspoon salt and ¼ teaspoon poultry seasoning, and cook until lightly browned on all sides, about 3-5 minutes, stirring frequently to avoid burning. Set aside.
To the same pan (no need to clean it in between), add 1 tablespoon butter and 1 tablespoon olive oil. Turn to medium heat, then add the onion, celery, carrot, ½ teaspoon salt, pepper, and 1 teaspoon poultry seasoning. Cook until just softened, about 3-5 minutes, stirring occasionally.
Sprinkle with flour and cook for 1 minute, stirring constantly. Add 2 cups of chicken stock, turn the heat to high, and bring to a rapid boil for 1 minute, stirring constantly. Turn off the heat and add the half-and-half, peas, and 1 teaspoon salt, stirring to combine.
Prepare a 9x13 ceramic baking dish with oil or cooking spray. Add the chicken, then pour over the vegetable mixture, stirring until well combined. Smooth to flatten the top, then cover evenly with the prepared bread cubes. Pour the remaining ¼ chicken stock over the bread cubes.
Cover tightly with foil, then bake for 30 minutes, or until bubbling around the edges. Uncover and bake for an additional 5 minutes, or until the bread cubes are golden brown.
Allow to set for 5-10 minutes, sprinkle with parsley if using, and serve.