The Bombay Sandwich with Cheese is the Mumbai vegetable sandwich with a processed cheese slice added to the stack. That one addition changes the sandwich more than its size suggests: the cheese melts under the press into the layers, binding the otherwise loose vegetables together and adding a salty, fatty smoothness that rounds off the sharp chutney and the dry chaat masala. The angle is richness and cohesion. A standard build can be bright and a little disjointed; the cheese slice glues it and softens its edges.
The build follows the familiar order with one extra step. The bread is spread with green chutney, then layered with the usual vegetables and dusted with chaat masala, and a processed cheese slice is laid in, often near the middle so it melts toward both faces, or sometimes against the bread so the inner crumb goes rich. The closed sandwich is buttered and grilled on the press until the outside colors and, critically, until the cheese has actually melted and started to fuse the layers. Good execution means the cheese fully softened and spread through the stack rather than sitting as a cold rectangle, with the chutney and masala still assertive enough to cut the added fat. Sloppy execution is a press pulled too early so the slice stays firm and rubbery and contributes nothing, or so much cheese that the sandwich turns one-note and the vegetables and chutney disappear under it. It is cut hot, when the cheese is still molten and stretches slightly at the seam.
Variation is mostly about the cheese: a single slice for a subtle bind, a double slice for a frankly indulgent version, or a swipe of cheese spread instead of a slice for a more even melt. This is a distinct member of the Bombay Sandwich family and not a default, and the plain pressed version, the heavily buttered crisp-shell version, and the non-grilled toast version each deserve their own article rather than being crowded in here. What defines the cheese version specifically is the interaction of melted processed cheese with the chutney-and-masala base: the cheese has to actually melt and the spicing has to stay loud enough to keep it from going flat, and a version where either fails collapses into a bland, greasy stack that misses the entire reason for adding cheese in the first place.