Start by melting the butter in a large saucepan over medium heat. Add the flour and stir constantly until the mixture turns golden brown, which should take about 5 minutes.
Gradually whisk in the beef broth until the mixture is smooth.
Add the cooked beef, onion, celery, carrot, parsley, salt, and black pepper. Stir everything together until the mixture is well combined.
Cook the mixture over medium heat, stirring constantly, until it becomes thick and bubbly, which should take about 10 minutes.
Remove the mixture from the heat and let it cool to room temperature. Once it has cooled, cover it with plastic wrap and refrigerate for at least 2 hours, or until it is firm.
Beat the eggs in a shallow dish. Place the breadcrumbs in another shallow dish.
Using a spoon, scoop out small balls of the chilled mixture and roll them into balls. Dip each ball into the beaten eggs, then coat it in the breadcrumbs.
Heat the vegetable oil in a deep frying pan over medium heat. Once the oil is hot, fry the balls in batches until they are golden brown, which should take about 2-3 minutes.
Remove the balls from the oil with a slotted spoon and drain them on paper towels.
