For the raw apple beet slaw: Shred the apple and beets in a food processor fitted with the grater attachment and transfer to a medium bowl. Toss with the oil, dill, vinegar, salt and pepper. Refrigerate and let the flavors meld while you make the rest of the components.
For the horseradish-pickle mustard: Combine the mustard, horseradish and pickle in a small bowl.
For the latke bun: Peel the potatoes and onion and shred them in a food processor fitted with the grater attachment; transfer to a dish towel-lined strainer fitted over a large bowl. Let sit for 5 minutes, and then squeeze the veggies using the towel to get out the excess moisture. Transfer to a large bowl. Add the matzo meal, egg and a pinch each of salt and pepper and use your hands to mix them together.
Fill a large cast-iron skillet with 1/2 inch oil. Heat over medium-high heat until the oil is very hot but not smoking. Working in batches, scoop the potato mixture by 1/3-cupfuls and form a bun shape. Carefully add the latkes to the skillet, flattening each slightly with a spatula. Fry, turning the latkes once, until golden brown and cooked through, about 3 minutes per side. Drain on paper towels and season with salt and pepper. Remove any loose bits of potato mixture from the oil in the pan between batches with a slotted spoon. You should have 12 bun halves.
For the corned beef burger: Pulse the corned beef in a food processor fitted with the blade attachment until a coarse burger-like consistency. Add the egg and matzo meal and pulse just to combine. Form into 6 burger patties (about 1/4 cup each). Add enough oil to a large cast-iron skillet to coat the pan and heat over over medium heat. Working in batches, cook the burgers until the outsides are brown and crispy, about 4 minutes per side. The insides will still feel on the rare side, and that's ok...and delicious.
To build Schmear 6 latke buns with applesauce. Top each with a corned-beef burger, then a pile of slaw. Schmear some horseradish-pickle mustard on the remaining 6 latkes, and place on top of the slaw. Cut your burger in half if you can't manage...it's a mouthful!