Lay the bacon in a cold skillet, turn the heat to medium and cook 8 to 10 minutes, turning, until crisp. Drain on paper towels and pour off all but a spoon of fat.
Split the rolls, butter the cut sides and toast them face down in the pan for 1 minute. Set aside.
Beat the eggs with the milk, salt and pepper. Pour into the pan over medium-low heat and push gently with a spatula until just set, about 2 minutes.
Divide the eggs into two roll-sized piles, lay 2 slices of cheese on each, cover the pan for 30 seconds so the cheese melts into the eggs.
Slide each pile onto a roll, add the bacon, salt, pepper and ketchup or hot sauce. Wrap tightly in foil for 1 minute so it steams together, then eat.
Tips
Wrapping in foil is not optional: it is what makes a bodega sandwich soft and melted.
Swap the bacon for griddled sliced corned beef or mortadella.