The three-way roast beef is a sandwich defined by a counting word, and the count is a condiment order, not an ingredient list. On the Massachusetts North Shore, a roast beef is built on rare beef sliced to translucence on a seeded onion roll, and the three things being counted are the dressing: barbecue sauce, mayonnaise, and a slice of cheese. Order it "three way" and you get all three. The grammar is the whole point. The beef and the roll are assumed; what the counter is actually asking you is which of the three accents you want, and the regional default answer is all of them at once.
The craft is in how those three counted elements divide the labor against a cold pile of beef. The barbecue sauce on the North Shore is a specific kind: a thin, sweet-sharp James River style, closer to ketchup-and-vinegar than to a thick smoky glaze, so it threads into the folds of the meat and supplies acid without coating it into a paste. The mayonnaise is the fat, spread to carry richness the lean rare beef does not have on its own. The cheese, usually American, is doing structural work as much as flavor work, half-melting against the warm-edged pile so it tacks the loose mound to the roll instead of letting it slide. The order language extends down from there with the same logic: a two-way drops one element, almost always the cheese, and a one-way is sauce alone, the build stripped to the single accent the eater cares about. The seeded onion roll is the fixed structural choice under every count, sturdy and faintly sweet so it carries a heavy cold pile and a wet sauce without the dipping a lesser roll would need.
The variations are mostly the counts themselves and the shops that codified them, and they run alongside the wider thin-sliced roast beef family rather than into it. The dipless North Shore build as a whole, the super beef that scales the same order up, the chain roast beef that flattened the idea for a drive-through, the Chicago Italian beef and Los Angeles French dip that answer the same lean-beef problem with jus instead of a counted dressing: each is its own discipline with its own rules and deserves its own article rather than being crowded in here.