Cut the peeled carrot and broccoli into easy to eat pieces, and boil in salted water. Moisten the panko in milk.
Chop the onion finely and put into a heatproof bowl. Mix with 1 tablespoon of vegetable oil. Do not cover with plastic wrap; microwave for 4 minutes.
When the onion has cooled down, add the rest of the hamburger ingredients into the bowl and mix well with your hands until sticky.
Form into patties of the size you like, and pan fry in 1 tablespoon of oil until browned on both sides. You don't need to cook the patties through at this stage.
If you can't fit all the patties into the frying pan at once, fry them in 2 batches. Reserve the juices that come out of the meat from the first batch, and return to the pan after the 2nd batch is done.
When all the patties are browned, add the drained mushrooms and butter.
When the butter has melted, add all the simmering sauce ingredients, and cook over low to medium heat while stirring occasionally until the sauce has thickened. (About 30 minutes.)
Transfer to a serving dish, add the broccoli and carrot, pour any sauce remaining in the pan over, and it's done.
Optionally drizzle on some fresh cream or coffee cream. Serve with bread on the side to dip into the sauce - delicious!
Sweet onions are very watery and can make the burgers fall apart, so use regular onions. Use ground meat with some fat in it, rather than very lean meat.
If you add the sauce ingredients all at once in Step 7 the temperature will cool too fast, so after adding the demi-glace sauce add the water and red wine little by little.
Story Behind this Recipe
My family loves hamburger steak simmered in sauce. If you use demi-glace in the simmering sauce, you'll have a restaurant-quality dish! It looks like it takes a lot of effort, but it's actually quite easy.
If you buy cheap off-cuts at a butcher or supermarket and ask "Can you grind it up for me?", they will usually do it for you for free. Freshly ground meat is fresher than pre-ground of course, and it looks and tastes better. This is a lot better than buying iffy pre-ground beef!