For the boxty pancakes: Preheat the oven to 200 degrees F. Grate 5 of the potatoes on the largest holes of a box grater. Toss with the vinegar and squeeze out the excess water between paper towels.
Cut the remaining 3 potatoes into 6 pieces each. Place them in a saucepan with the garlic and cover with water by about 1 inch, sprinkling generously with salt. Bring to a boil, reduce to a simmer and cook until the potatoes are fork tender, 10 to 12 minutes. Drain. While still hot, pass the potatoes through a food mill or mash with a potato masher.
Combine the grated potatoes with the mashed potatoes. Beat the eggs and add into the potatoes. Sprinkle with salt.
Coat a large saute or cast-iron pan with olive oil over medium-high heat. Make a 1 1/2-inch patty and cook until brown and crispy, 3 to 4 minutes per side. Eat the first pancake to make sure the potato mixture is seasoned perfectly. Season again, if needed.
Working in batches, make and cook all the pancakes, blotting on paper towels and seasoning with salt. Hold the pancakes on a sheet tray in the oven to keep warm until serving.
For the lamb loin with whole-grain mustard aioli: Sprinkle the lamb with salt and brush with whole-grain mustard. Coat a large saute pan with oil over medium-high heat. When the oil starts to shimmer, add the lamb and sear on all sides, being careful not to burn the brown bits on the bottom of the pan. Cook until a thermometer reads 135 degrees F when inserted into the thickest part of the loins. Transfer to a small sheet tray and tent with aluminum foil. Allow to rest at least 10 minutes.
Combine the mustard, mayonnaise, chives and parsley in a medium glass mixing bowl.
Thinly slice the lamb. Place a dollop of the aioli on top of the boxty, and top with a slice of lamb and another dollop of the aioli.