Melt butter in a small Dutch oven over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add milk; cook over medium heat, stirring constantly, until thickened and bubbly. Stir in salt, celery salt, and chicken; continue to cook until thoroughly heated. Stir in chopped egg.
Spoon chicken mixture over Crunchy Corn Sticks; sprinkle with paprika. Serve immediately.