In a large mixing bowl, combine chilled ground beef with a generous pinch of salt and pepper. Gently mix until just combined to keep them tender, then divide the mixture into 12 equal portions. Shape each portion into small patties, making a slight dimple in the center of each.

Whisk together mayonnaise, minced garlic, Dijon mustard, and lemon juice in a medium bowl until smooth. Season with salt and pepper to taste, then refrigerate for at least 15 minutes.

Heat a large skillet over medium-high heat and add a small drizzle of oil. Cook the patties for about 3-4 minutes on each side, or until browned and internal temperature reaches 160°F.

Slice the mini brioche buns in half and toast the cut sides in the same skillet for about 30 seconds until golden brown.

Spread a layer of garlic aioli on the bottom half of each bun. Layer mixed lettuce, a cooked patty, a slice of cheddar cheese, and quick-pickled cucumber slices. Top with the bun lid and serve warm.