Add beef, croutons, chives, red onion, beaten egg with water, salt, and pepper in a large bowl. Mix thoroughly with your hands and form 2 patties, about 1.5-inch thick.
In a large non-stick pan, add your vegetable oil over medium heat. Cook hamburgers on each side for 3 to 4 minutes.
To keep the juices in, do not move or flatten patties. Set aside on a plate and cover with foil while you prepare your mushroom red wine reduction.
In the same skillet, add your cremini mushrooms and cook, turning frequently (adding more oil if necessary) until nicely browned. Add the cornstarch to the beef broth and stir thoroughly. Pour over mushrooms, add the red wine and continue stirring until thickened, about 5 minutes.
Spoon over hamburger patties and serve immediately.