The thick-omelette tamago sando replaces the egg salad with a single warm slab of atsuyaki tamago, and that one swap changes everything about how the sandwich eats. Instead of a cool mash spread edge to edge, you get a fat rectangular block of folded Japanese omelette wedged between soft shokupan, its layered cross section visible where the sandwich is cut. Atsuyaki tamago is the sweet-savory rolled omelette built up in a rectangular pan one thin layer at a time, each pour rolled back onto the last until the whole thing is dense, springy, and faintly sweet. Put that between bread and the result is bouncy rather than creamy, eggy in a clean cooked way rather than rich with mayonnaise, and substantial enough to feel like a small meal.
The craft is in the omelette long before it ever meets bread. Beaten egg is seasoned with dashi or sugar and soy, sometimes mirin, and cooked in repeated thin layers that are rolled and built into a tall block, which is what gives the finished slab its characteristic concentric structure and tender bounce. It has to be cooked through but not dried out, set firmly enough to hold its shape as a clean rectangle yet still moist when bitten. The shokupan is trimmed and left soft, usually with a thin layer of kewpie mayonnaise or a smear of karashi-spiked mayo on the bread to lubricate the bite and sharpen the omelette's sweetness. A good one shows a thick, evenly layered slab that fills the bread corner to corner, warm or at least not fridge-cold, the egg springy and just sweet enough to register without turning into dessert. A sloppy one is easy to spot: an omelette cooked unevenly so one end is dry and the other underset, a slab too thin to justify the form, sweetness pushed so far it cloys, or bread left bare so the dense egg eats dry. The bind is structural in a different way from the salad version, since the omelette is a solid object; the mayonnaise and the give of the bread do the work of gluing the slab in place so it does not slide free on the first bite.
That makes this a close sibling of the Kansai and Kyoto schools, where a dashimaki omelette stands in for egg salad as a matter of regional habit rather than novelty. The variations move along the omelette's seasoning and heft: some lean hard into dashi for a savory, almost soup-scented egg, some push sugar for a dessert-leaning sweetness, some stack the slab against a layer of mayonnaise-bound salad to get both textures in one bite. The double-egg build that does exactly that, and the soft-scrambled version that runs in the opposite direction toward a loose curd, are each their own technique with their own logic. Each deserves its own article rather than being crowded in here.