Slice the chilled steak thinly across the grain.
Melt half the butter in a skillet over medium heat and turn the steak with 1/4 teaspoon salt for 3 minutes until reheated to 165°F.
Transfer the steak to a plate and whisk the eggs with the remaining salt and black pepper.
Add the remaining butter and stir the eggs over medium-low heat until soft curds form and reach 160°F.
Fold the steak into the eggs, divide the mixture into four piles and cover each with provolone for 30 seconds.
Load the hoagie rolls and serve while the eggs remain soft.
