Preparation

Brine the pork

• Fill a large pot with 1-1/2 gallons of water. Add the salt, sugar, allspice, pepper, cloves, juniper berries, carrot, onion, thyme, parsley, and bay leaves, and bring to a boil. Remove from the heat and allow to cool completely. Submerge the pork in the brine. Cover tightly and refrigerate for up to 2 days.

Make the aïoli

• Put the yolks, lemon juice, mustard, and garlic in a blender and pulse until well combined, about 10 seconds. With the machine running, slowly drizzle in the olive oil and continue to process until the mixture becomes thick, about 2 minutes. Season to taste with salt and pepper. Refrigerate in a covered container until ready to serve.

Make and serve the burgers

• Remove the pork from the brine, rinse, and pat dry. Finely chop the pork and then pass through a meat grinder. (If you don’t have a meat grinder, transfer the finely diced pork to the bowl of a stand mixer fitted with the
  paddle attachment, and mix on medium speed to break it into smaller pieces, 3 to 5 minutes.) Form the mixture into six 6-oz. patties the diameter of the bun; reserve any remaining pork for another use.
• In a large skillet, preferably cast iron, heat the oil over medium heat. Add the onion, 1/2 tsp. salt, and 1/4 tsp. pepper, and cook until softened and golden brown, about 8 minutes. Transfer to a bowl. In the same skillet and working in batches, cook the patties over medium-high heat until golden brown, about 6 minutes, then flip and cook until golden on the second side, about 5 minutes. Top the burgers with the onions and cheese, and cover to let cheese melt, about 1 minute.
• Transfer the burgers to a platter, and tent with foil to keep warm. Wipe out the skillet, and then return to medium-low heat. Toast the buns in the skillet. Spread mustard on the bottom of each bun and aïoli on the top. Pile shredded lettuce on the bottom of each bun, and then top with a burger and sliced tomato. Cover with the top bun, and serve.