Bring the broth to a bare simmer in a wide skillet and lower the roast beef into it for 2 minutes until hot but not curled.
Melt the butter in a saucepan, whisk in the evaporated milk and cornstarch, and simmer for 1 minute.
Remove the saucepan from the heat and stir in the cheddar by handfuls until the sauce flows smoothly.
Mix the ketchup and vinegar in a small bowl and spread a thin layer on the toasted cut faces of the rolls.
Drain the warm beef well, pile it onto the 4 rolls, spoon cheese sauce over it and close the sandwiches.