Peel and grate the potatoes. Submerge the grated potatoes in ice water, strain, and squeeze out as much liquid as possible. In a bowl, mix the grated potatoes with an egg, salt, and potato starch until well combined. Freeze the mixture for an hour to firm up. Shape into 4.5x4.5-inch squares, ½-inch thick. Set in the fridge for an hour to firm up.

Make the chipotle mayo: blend mayonnaise, chipotles, adobo sauce, lime juice, and a pinch of salt until smooth. Optional: Instead of blending, finely chop the chipotles and mix everything together.

Make the quick pickled jalapeños: in a heat-proof jar, combine sliced jalapeños, lime slices, and garlic. In a small saucepan, combine white vinegar, water, sugar, and salt. Bring to a simmer over medium heat, stirring until the sugar and salt dissolve. Pour the hot mixture over the sliced jalapeños in a heatproof jar. Let sit for at least 15 minutes before using, but the longer these sit the better.

Prepare the egg and cheese: in a bowl, beat the eggs with water, salt, and pepper. In a skillet over medium heat, melt butter and add the eggs. Gently scramble until 80% cooked. Turn the heat off, add the American cheese in the center, and let sit for 2 minutes. Fold the eggs into a square shape, ensuring the cheese stays in the center.

Assemble the sandwich: Toast the deli rolls. Spread chipotle mayo and 1-2 teaspoons of ketchup on both halves of the rolls. Add the egg and cheese square, followed by the crispy hashbrown. Top with desired amount of the jalapeños. Close the sandwich and serve immediately.