Prepare fries by cutting potatoes into 10 mm (3/8 inch) sticks. Rinse in cold water until runoff is clear to reduce surface starch, then dry thoroughly on towels.
Par-fry potatoes at 160 C (320 F) for 4-5 minutes until soft but pale. Drain and cool on a rack; this two-stage frying yields a crisp exterior when finished at higher heat.
Form four patties of about 200 g (7 oz) each, gently shaping without overworking the meat. Season both sides with 5 g (1 tsp) salt and 2 g (1/2 tsp) pepper just before cooking to avoid drawing out moisture.
Heat a cast-iron skillet over medium-high until smoking point approaches. Add a thin film of oil. Sear patties 3 minutes per side for medium-rare, pressing slightly once in the first 30 seconds to ensure contact, then add cheese and cover for 30 seconds to melt.
Increase fryer oil to 190 C (375 F). Finish fries in hot oil for 2-3 minutes until deep golden and crisp. Drain on paper and immediately season with salt to stick to the hot surface.
Toast buns in butter in the same skillet for 30-45 seconds until golden, which adds a nutty aroma and prevents sogginess from burger juices.
Assemble: spread a tablespoon of sauce on each bun base, place the patty, add crisp lettuce or pickles if desired, and cap with the bun top. Serve fries alongside while both components are hot for optimal texture.