Melt the butter in a saucepan. When hot and bubbly, and the onion, carrot and celery. Cook until the vegetables begin to soften. Add the beef, potato and mushrooms and cook for an additional 4 minutes. Transfer the mixture to a bowl, mix in the rice, sprinkle with the flour and then with the broth. Mix in the egg. Sprinkle with salt and pepper.
Lightly butter a baking sheet. Using a lightly oiled bowl for a mold, pack about 1 1/2 cups of the mixture into a lightly oiled small bowl and turn out onto the baking dish. Repeat 3 times for 4 hash cakes.
Alternatively, these can be hand shaped. Lightly dusted with additional flour and sauteed in your grandmother's heirloom cast iron skillet.