Although sautéed and stir-fried dishes do not figure prominently in Japanese cuisine, simple dishes in that category will appear on the tables of most Japanese homes.
The scale, however, is much more diminutive than most Chinese, and certainly most Americans, would expect.
Historically, oil was fairly expensive in Japan, and a typical farming family might have gotten away with a single modestly-sized bottle of vegetable oil over the course of an entire year, even in the era of cast-iron pots. A tiny bottle of toasted sesame oil, mostly used a few drops at a time, might provide a flavor boost to otherwise simple dishes.
Itamemono, or pan-sautéed dishes, generally have a fairly subtle flavor. Even soy sauce is used with a very light hand. Dishes do not acquire the "red" color of Chinese-style stir-fried dishes.
This type of dish is best with a short list of ingredients, prepared in small batches; I don't think I made this with much more than 1 to 1 1/2 cups of raw ingredients, and it was enough for two or three Japanese servings along with other dishes. My tiniest omelet pan did the trick. For a group of four or five people, you could get away with making a larger quantity in a 10" skillet.
I used some snow peas (sayaendō), carrots prepared with a simple rolling cut, and onions. I dunked some abura-age (tofu puffs, perhaps) in very hot water and squeezed the water out to prepare it for the pan. This helps the aburaage more readily absorb salt and seasonings, and coincidentally slightly reduces the oil content.
This kind of sauté is done with very little oil at a fairly high temperature. I add a small pinch of salt every time I put a new ingredient in the pan, then finish with some soy sauce, sake, and mirin.
Japan doesn't have the French convention of caramelizing onions, but if you bring the onions just past translucent they'll add a great aroma and natural complexity to the flavor of the dish. Add the carrots, cook for a minute or two longer, then add the snow peas and aburaage. Once these are a bit shiny, add a small splash of soy sauce, a good tablespoon of sake, and maybe even a little dashijiru, then perhaps one or two drops sesame oil. In some cases you may want to add a touch of sugar or mirin, but I think the onions and carrots are naturally sweet enough that additional sugar is usually unnecessary to achieve an ama-karai (sweet-salty) taste.
Simmer briefly, taste, adjust seasonings if needed, and serve in small bowls.