Bring rice and 1 1/4 cups water to a boil in a saucepan. Cover, reduce heat to low and cook for 15 minutes; rest off heat, covered, for 10 minutes.
Toss cucumber with vinegar and salt and set aside. Mix soy sauce, brown sugar, cornstarch and the remaining 1/4 cup water until smooth.
Heat oil in a large skillet over medium-high heat. Press beef into a thin layer and sear undisturbed for 3 minutes. Turn in broad pieces and cook for another 3 minutes, then break into coarse crumbles and cook to 160°F.
Drain excess fat, leaving about 1 tablespoon. Add ginger, garlic and scallion whites and stir for 1 minute.
Stir the sauce again and pour it over the beef. Bubble for 1 to 2 minutes until glossy and lightly thickened, then fold in scallion greens for 30 seconds.
Fluff rice and divide among four bowls. Spoon beef over the rice and add drained cucumber alongside it.