Step 1

Whisk ¼ cup distilled white vinegar, 2 tsp. sugar, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt in a small bowl until sugar and salt are dissolved. Add 2 medium shallots or 1 small red onion, thinly sliced, and let sit, tossing once or twice, until ready to serve.

Step 2

Mix together ½ cup mayonnaise, 3 Tbsp. Sriracha, and 1 tsp. smoked paprika in a small bowl; season with salt. Set aside.

Step 3

Toast 2 tsp. cumin seeds and 2 tsp. fennel seeds in a dry large cast-iron skillet over medium-high heat, stirring often, until fragrant, about 30 seconds. Transfer to spice mill or mortar and pestle and let cool slightly, then coarsely grind. Transfer to a large bowl; reserve skillet.

Step 4

Add 1 lb. ground lamb, 2 garlic cloves, finely grated, remaining 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and remaining 1 tsp. smoked paprika to bowl with spices; season with freshly ground black pepper. Using clean hands, mix until evenly incorporated. Divide mixture into 6 portions and form into loosely packed balls.

Step 5

Heat reserved skillet over medium-high until you can see wisps of smoke. Add 1 Tbsp. vegetable oil, then 3 balls of meat, spacing evenly apart. Firmly smash down meat as thin as possible with a heavy metal spatula (you can also use a slightly smaller heavy skillet to weigh down patties as they cook); season with salt and pepper. Cook patties, undisturbed, until deeply browned underneath, about 2 minutes. Flip and cook until cooked through, about 1 minute. Transfer to a plate. Repeat with remaining balls of meat, adding more oil if skillet looks dry.

Step 6

Pour off fat from skillet; discard. Wipe out skillet, then pour in 1 tsp. vegetable oil and return to medium heat. Working in 2 batches and adding more oil if pan looks dry, toast 6 brioche buns, split, cut side down, until golden brown, about 1 minute. Transfer to a work surface.

Step 7

Heat remaining 2 Tbsp. vegetable oil in a large nonstick skillet over medium. Crack 3 large eggs into skillet and cook until whites are just set, about 4 minutes; transfer to a plate and season with salt and pepper. Repeat with remaining 3 large eggs.

Step 8

Spread cut sides of buns with reserved Sriracha mayo. Build burgers with patties, eggs, some drained pickled shallots, and a few arugula leaves.