Melt 2 Tbsp (30 ml) of butter in the microwave or a small pan.
Brush the steaks with the melted butter and season with pepper; let the steaks come to room temperature.
Peel and slice the potatoes into ¼ inch (0.64 cm) thick sticks. Soak in ice water for 15 – 20 minutes.
In a large pot, heat 2 inches (5 cm) of vegetable oil to 325°F (165°C).
Drain the potatoes from the water and dry them completely.
Fry potatoes in batches until cooked through, but not brown, about 4-6 minutes. Transfer to a paper towel lined tray to drain. Heat oil to 375°F (190°C).
Heat the grill pan over medium heat.
Season steak with salt and add to the grill pan once the pan begins to smoke slightly. Cook steaks for 4 minutes per side for medium rare.
Refry potatoes until crisp and golden brown, about 1-2 minutes per batch. Transfer to fresh paper towel lined baking sheet to drain.
Divide remaining butter evenly and top on each steak. Top each steak with a handful of frites and serve mayonnaise on the side.