Melt butter in a saucepan over medium heat. Whisk flour into melted butter; cook and stir until raw flour taste cooks off and a paste forms, about 1 minute. Slowly whisk milk and chicken broth into butter mixture, whisking constantly, until mixture is thick and smooth, 5 to 10 minutes. Remove sauce from heat and cool slightly, about 10 minutes.
Stir chicken, 1 cup bread crumbs, eggs, onion, parsley, garlic powder, celery seed, salt, and black pepper together in a bowl; add cooled sauce to chicken mixture and stir until well-combined. Cover the bowl with plastic wrap and refrigerate until chilled and firm, at least 2 hours.
Shape chilled chicken mixture into 6 patties. Pour remaining bread crumbs into a shallow bowl and roll each chicken patty in bread crumbs to coat completely.
Heat oil in a large skillet over medium-high heat. Cook patties in hot oil, turning once, until patties are cooked through and golden brown on each side, about 10 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
