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The trickiest thing about this dish is just timing everything right. I started the beef and cucumbers, and then started the rice so it could cook while the beef was marinating and the cukes were pickling. You could also start the rice first.
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For the ground beef, mix everything together and then place in the fridge in a sealer container for 30 minutes.
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In the meantime begin the cucumbers by heating the sugar and water in a small pot. Stir until the sugar is dissolved. Then remove from the heat and add the pinch of salt and the vinegar. Let cool.
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Keep an eye on it--when it is cool enough, pour it over the sliced cucumbers.
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If you have not started the rice by now, start it now.
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In the meantime, heat a large nonstick skillet or wok over medium high heat. Add the 2 teaspoons of vegetable oil, and when it is hot add the mushrooms. Cook until they have released some of their water and are starting to stick, at which point add the garlic and the mirin. Stir and use the mirin to deglaze the bottom of the pan.
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When the mushrooms have mostly cooked down, add the soy sauce and stir. Let cook another minute, before stirring in the sesame oil and transferring the mushrooms to a bowl.
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Wipe out the pan if it has anything stuck in it and return it to medium high heat. Add the bean sprouts and toss to heat for 1-2 minutes. Then add the rice vinegar and toss another 2 minutes. Then transfer to a bowl.
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Place the seasoned ground beef into the pan over medium heat. While it is cooking, prepare the daikon radishes.
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Massage all of the ingredients for the daikon radish salad together and then transfer to a bowl.
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When the ground beef is browned all the way through, transfer it to a bowl--if there is a lot of excess oil, you can drain it off first.
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Wipe the pan out. Assemble each bowl by placing rice at the bottom, the ground beef on top of the rice, and then the vegetables arranged over that, including the kimchi and gochujang. Fry one egg per bowl to place on top--preferably with a runny yolk, as the yolk will meld with the gochujang to create a delicious sauce.
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Serve immediately.