Slice the dill pickles first so they’re ready to go. Cut each pickle into 4 long spears, then cut each spear in half crosswise. Slice each short spear lengthwise into 3 to 4 thin strips, creating narrow pickle matchsticks. Set aside.

If using cheese, stack the two slices of American cheese on top of one another on a cutting board. Cut the stack into 4 even strips, then cut each strip into 3 equal pieces to make 12 small stacked squares. Keep these chilled or off to the side until you’re ready to stuff the latkes.

Fit a food processor with the coarse grating disc. Grate the onion wedges first, then grate the potatoes directly on top of the onion. (Alternatively, use the large holes of a box grater.) Transfer the grated onion and potato mixture to the center of a clean kitchen towel and add the sliced pickles on top. Gather up the corners of the towel and, working over a large, clean bowl, twist and squeeze as hard as you can to remove as much liquid as possible. The drier the mixture, the crisper the latkes will be.

In a separate large mixing bowl, whisk the egg until it is fully blended. Add the squeezed potato-onion-pickle mixture to the egg. Look into the bowl that caught the squeezed-out liquid: gently pour off and discard the liquid, reserving any white starch that has settled at the bottom. Scrape this starch into the egg bowl—it will help bind the latkes.

Sprinkle the matzo meal, toasted sesame seeds, kosher salt, and paprika evenly over the potato mixture. Using a rubber spatula, fold everything together until the potato strands are evenly coated and the mixture feels cohesive but not overly wet.

Line a baking sheet with parchment paper. Using a 1/4-cup measuring cup, portion the latke mixture into 12 mounds on the sheet. If you’re making cheese-stuffed latkes, work with one mound at a time: split it into two equal pieces, press one half into a roughly 3" round patty, place one piece of the stacked American cheese in the center, then cover with the second half of the mixture. Flatten and shape into a 3 1/2" patty, making sure the cheese is fully enclosed. If you’re skipping the cheese, simply press each portion into a 3 1/2" flat patty.

Set a wire rack inside a rimmed baking sheet and place it near the stove for draining the fried latkes. In a large, preferably cast-iron skillet, pour in enough neutral oil to reach a depth of about 1/4". Heat the oil over medium-high heat until it reaches 350°F on a deep-fry or instant-read thermometer. If you don’t have a thermometer, drop in a tiny bit of latke batter—if it sizzles immediately and begins to brown, the oil is ready.

Working in batches of 3 or 4 latkes at a time to avoid crowding, carefully slide the patties into the hot oil using a metal fish spatula or similar flat spatula. Fry without moving them for 3 to 4 minutes, or until the bottoms are deeply golden and crisp. Gently flip and fry the second side for another 2 to 3 minutes, until equally golden and cooked through.

Transfer the fried latkes to the prepared wire rack to drain excess oil. While they’re still hot, sprinkle lightly with a bit more kosher salt. Repeat the frying process with the remaining patties, adjusting the heat as needed to maintain a steady sizzle and prevent burning.

To serve, arrange the hot latkes on a platter. Drizzle generously with Big Mac sauce and scatter shredded iceberg lettuce alongside or over the top for a burger-style finish. Serve immediately while the edges are crisp and, if stuffed, the cheese inside is still melty.