In a saucepan over medium heat, melt the butter. Add the flour and stir to combine. Add the broth and bring to a boil, stirring with a whisk. Reduce the heat and simmer for about 5 minutes. Remove from heat. Gradually pour in the egg yolk mixture, whisking constantly. Reheat while stirring until the sauce thickens. Do not let the sauce boil. Add the hard-boiled eggs. Adjust the seasoning.