Pork is ideal here because it doesn't take long to become sublimely tender and because its melting fat provides an added dimension. But a super-fatty piece of pork will make the dish taste greasy. The best cuts are shoulder, trimmed of excess surface fat, or so-called country-style ribs, the meat stripped from the bone.
Put oil in a large, deep skillet and turn heat to high. A minute later, add pork and cook, undisturbed, until nicely browned on one side, 3 to 5 minutes. Shake the pan or use tongs to turn, then toss in the ginger and garlic.
A minute later, add soy sauce, sugar and 1/2 cup water, along with mirin, honey or sugar. Turn heat to low, cover and simmer about 30 minutes, until pork is fairly tender. Add shiitakes, stir, and cook about 10 more minutes, until pork is quite tender.
Uncover, raise heat to high, and cook, turning pieces frequently, until liquid is reduced to a saucy consistency. Discard ginger and garlic if you like, and serve over white rice.