A jammy yolk, caught at the moment it stops being liquid but has not yet gone solid, is the whole reason the half-boiled tamago sando exists. This is the modern, photogenic reading of the egg sandwich: instead of fully set egg mashed into salad, soft-boiled eggs with deep orange centers are laid between slices of shokupan so that the cut face shows a row of glossy yolks, some of which run a little when the sandwich is halved. It is a deliberate departure from the cool, uniform konbini standard, built to look molten and eat richer.
The technique is a timing problem before it is anything else. Eggs are boiled just long enough to firm the white while leaving the yolk thick and fudgy, then peeled, halved or arranged whole, and set into the bread often with a smear of kewpie and sometimes a little of the mashed-egg base underneath for stability. The defining quality is the yolk: it should be dense and custardy, on the edge of flowing, deep in color, with the kewpie sharpness cutting what would otherwise be a very rich mouthful. A good half-boiled sando shows clean intact yolks at the cut, a white that is set but tender rather than rubbery, and just enough binder to hold the eggs without burying them. The faults are unforgiving because the margin is thin. Boil a minute long and the yolk sets up chalky and the entire point is gone; boil short and the white stays slippery and the egg will not hold a cut; under-support the eggs and they roll out of the sandwich on the first bite; let a truly runny yolk sit against soft shokupan and it soaks the crumb to mush before it reaches the eater. The bind is unusually delicate here, because it has to cradle whole soft eggs without crushing them and contain a yolk that wants to escape.
This is the contemporary branch of the family, defined by a yolk state rather than a region. It departs from the Kanto style's fully set, finely mashed homogeneous salad and from the Kansai and Kyoto omelette pole, where a thick dashimaki makes the sandwich bouncy and savory instead of molten. The double-egg build chases indulgence by stacking salad and omelette together, and the demi-glace treatment goes Western with a dark poured sauce over the egg. Each of those is a separate technique with its own intent and deserves its own article rather than being crowded in here.