The BLT with egg adds one component to a sandwich that was defined by having almost none, and the fried egg changes what the sandwich is for. Bacon, lettuce, tomato, mayonnaise, and a fried egg on toast is the whole build. The egg is not a garnish on a BLT; it is a second structural and flavor center that pulls a lunch sandwich toward breakfast, and the defining decision is how that egg is cooked, because a runny yolk and a hard yolk make two different sandwiches out of the same parts.
The craft is the BLT's craft with the egg as the new variable. The bacon is still the flavor and the spine, rendered crisp and laid flat so every bite gets some and the soft sandwich has a little rigidity. The tomato is still the threat, salted and walled off from the toast by mayonnaise spread edge to edge, because adding an egg does nothing to fix the original's race against the tomato's water. The lettuce is still the cold crunch the rest of the build lacks. The egg is the decision: cooked over-hard, it is a firm, neutral protein layer that adds body without endangering the structure; cooked over-easy, the broken yolk becomes a sauce that binds the bacon and the toast into something richer and far more fragile, which is why a runny build is eaten fast and often with a fork standing by. The egg is placed against the bacon rather than the tomato so its heat does not accelerate the tomato's collapse, and the toast still has to be crisp and recent because none of the additions buy the sandwich more time.
The variations stay inside the same frame, and each deserves its own article rather than being crowded in here. Add avocado for a BLAT and the richness shifts again; run it open-face under the yolk and it becomes a knife-and-fork plate; the club sandwich is the braced descendant that solves the instability this build, like the plain BLT, accepts on purpose. Each is a single deliberate move on a sandwich with very few parts, which is exactly why each one earns a name.