1. To prepare the potato component, place the scrubbed potatoes into a pot, covering them with cold water by about an inch. Add 2 tablespoons of kosher salt and bring to a boil over high heat, cooking until a skewer inserted into the center meets just a bit of resistance. Drain completely and allow them to cool. (Note: You can cook and refrigerate the potatoes up to 3 days ahead of time.)
2. In a large bowl, combine the vinegar, sugar, celery seeds, and mustard with a teaspoon of salt and a quarter-teaspoon of pepper, whisking until the sugar has completely dissolved. Add the oil and shredded cabbage and mix well until combined. Transfer to the refrigerator for at least 30 minutes, or up to 4 hours.
3. Prepare the grill for direct grilling over a medium-high fire (550-660°F) with an indirect cooking zone as well. A charcoal fire is preferred.
4. Slice the potatoes into ¼ inch thick slices, brush both sides liberally with oil and season with salt and pepper. Grill the slices over indirect direct heat until golden brown and crispy on both sides, 3 to 5 minutes per side. Remove to a sheet and season with a bit more salt.
While the grilled flank steak rests for 10 minutes after cooking according to the recipe, butter the cut side of each roll. To ensure even browning, place the rolls buttered-side down over direct heat and press firmly with a grill press until lightly golden. Transfer them cut-side up to the indirect cooking zone, crown each half with a slice of cheese, and close the grill lid for about 30 seconds to let the cheese melt.
Assemble each sandwich by layering the sliced steak, grilled potatoes, a generous heap of coleslaw, and fresh tomato slices onto the bottom rolls before capping with the top rolls. Serve remaining crispy potatoes on the side.
