Slice the chilled steak thinly across the grain.
Heat half the butter in a skillet over medium heat and turn the steak with 1/4 teaspoon salt for 3 to 4 minutes until reheated to 165°F, then transfer it to a plate.
Whisk the eggs with the remaining salt and black pepper.
Add the remaining butter to the skillet and stir the eggs slowly for 2 to 3 minutes until soft curds form but the surface still looks moist.
Divide the steak and eggs among the rolls, add one cheddar slice to each and close the sandwiches for 1 minute so the cheese softens.
Spread the top halves with mayonnaise and serve while the eggs remain tender.