Turn the heat to high under a large skillet that can later be covered. Add the beef cubes a few at a time and brown well, adjusting the heat so the meat browns but does not burn and turning beef to brown on all sides; this will take about 10 minutes. Remove the meat to a plate when it is done, and turn the heat to low.
Add garlic and ginger to the pan and cook, stirring occasionally, until the garlic colors, about 2 minutes. Add the water, then the soy sauce and either sugar or mirin. Add the meat and bring to a boil; cover, reduce heat, and simmer.
Adjust the heat so the mixture simmers steadily but not violently and cook, stirring occasionally, until the meat is tender, 50 minutes, depending on the cut you used (tenderloin cooks much faster than chuck). Add the kimchi, and cook 10 minutes or until heated through. (You can prepare the dish several hours in advance up to this point. Cover and set aside until you're ready to eat, then reheat; or cover and refrigerate overnight before reheating.)
Uncover, then taste and adjust seasoning. Stir in sesame oil, sprinkle with sesame seeds and scallions, and serve immediately over cooked rice.