In a large pot of heavily salted boiling water, cook potatoes until just-tender, about 10-15 minutes. You can check for doneness by piercing a potato with a fork. Or, well, by eating one. Drain.
When cool enough to handle, using the heel of your hand or a meat pounder, gently flatten each potato into a burst disc. Set aside on a small sheet pan or plate until ready to fry.
Divide the ground beef into 2 or 3 portions, and gently shape into patties. Season generously with sea salt.
In a skillet, combine shallot with 2 tablespoons cold ghee, then set over medium-low heat. Toast shallots until crisp and caramelized, about 5-7 minutes. Fish out crisped shallots as soon as they are done, and let drain on a paper towel lined plate.
Wipe out the skillet, then add another 2 tablespoons ghee. Turn the heat to medium-high, and sear the burgers until deeply browned, about 4-6 minutes per side.
Wipe out the skillet once more, and melt the remaining ghee. Place the potatoes in a single layer, and fry undisturbed until crispy, about 2-3 minutes per side. Transfer to a plate, and spoon browned butter on top. Shower with flaky salt, crisped shallots, and parsley.
Top buns with aioli, patties, tomatoes, and lettuce. Serve with crispy potatoes.