Breakfast Smash Burgers

Name: Breakfast Smash Burgers
Ingredients: ¼ cup distilled white vinegar; 2 tsp. sugar; 2 tsp. Diamond Crystal or 1 tsp. Morton kosher salt, divided, plus more; 2 medium shallots or 1 small red onion, thinly sliced; ½ cup mayonnaise (preferably Kewpie); 3 Tbsp. Sriracha; 2 tsp. smoked paprika, divided; 2 tsp. cumin seeds; 2 tsp. fennel seeds; 1 lb. ground lamb; 2 garlic cloves, finely grated; Freshly ground black pepper; 3 Tbsp. plus 1 tsp. (or more) vegetable oil, divided; 6 brioche buns, split; 6 large eggs; Arugula leaves (for serving); A spice mill or mortar and pestle