Preheat oven to 375ºF. Bring broth to a simmer in a medium saucepan. Add chicken and poach at a gentle simmer until no longer pink in the center (160ºF internal temperature), about 12 minutes. Transfer chicken to plate and let sit for 5 minutes. Cut into bite-size cubes. Reserve chicken and stock separately.
Meanwhile, heat the oil in a large deep sauté pan over medium heat. Add the onions, carrots, celery, and salt, and cook until vegetables soften, about 6 minutes. Add butter and stir until melted. Stir in flour until vegetables are coated with a thick paste. Cook for 2 minutes, scraping pan bottom often. Slowly pour in stock, stirring until mixture thickens, 2 minutes. Stir in the thyme and pepper.
Whisk together egg yolks and half-and-half in small bowl. Whisk 1/4 cup of hot filling mixture into the eggs; then whisk the egg mixture back into the filling. Cook over low heat for 1 minute. Fold in the chicken, peas, and parsley.
Transfer filling to a 13-inch-by-9-inch baking dish. Top with croissant cubes, filling holes with pieces of croissant to make a solid layer of croissant. Bake until croissants are nicely browned, 10 to 12 minutes. Let sit for 5 minutes before serving.
Tip: For extra-rich potpie, line the bottom of the baking dish with an additional four Panera croissants, split in half through the side and arranged in the dish cut-side up.