Combine the chicken, shredded carrot, garlic powder, celery seed, poultry seasoning, 1 teaspoon of salt, ½ teaspoon pepper, and breadcrumbs in a mixing bowl. Divide the mixture into four equal portions and form into balls.
In a large skillet, heat 2 tablespoons butter over medium heat. Add the chicken balls and, using a spatula, smash the balls into burger patties. Cook for about 3 minutes per side or until done. Remove from the skillet and keep warm. Add the remaining 2 tablespoons butter, let it melt, then whisk in the flour. Cook, whisking, for 1 minute. Add 1 cup milk and whisk. Cook until thickened, about 2 minutes. Add the peas, ¾ teaspoon salt, and ½ teaspoon pepper. Stir.
Place a burger on each of the bottom buns, top with the sauce, and then the top bun.