Mix ground beef, ketchup, hoisin sauce, beaten egg and bread crumbs in a large bowl. Works best when you use your hands. Try not to over mix. When all ingredients seem incorporated, form your patties. I find this easiest to do with a parchment lined sheet pan. 

Place beef patties covered with plastic wrap into the fridge while you make your sauce. 

Add butter to a pan on medium heat. When butter has melted, add your diced onion and sliced mushrooms. Cook mushrooms and onion for around 10 minutes until the mushrooms start to get soft. 

Add the two tablespoons of flour and stir in with the mushrooms and onion. When you can no longer see white flour (2 to 3 minutes) you can add the ketchup, hoisin sauce and beef broth. 

Raise the heat on the pan to high until the sauce starts to simmer. At that point turn the heat back down to medium heat and simmer for 10 minutes until the sauce is thick. Taste for seasoning and add salt and pepper if you think it needs some. 

At this point if you have a second pan, you can go ahead and start cooking the beef. If you do not have another pan, pour your sauce into a bowl and put it aside until the beef is done. 

Add your beef patties to a pan over medium high heat. Cook each patty 3 minutes per side.

Once your patties are cooked, add your sauce back to the pan with the patties until the sauce is warm. 

Add one patty and a bit of sauce to a toasted bun and enjoy.