Place the red onion rounds into a bowl along with the sugar and vinegar. Be sure to use enough sugar and vinegar that they are totally submerged. Drain and rinse with water before serving.
Meanwhile, in another bowl, combine the finely chopped mint, lemon zest and juice, and the Greek yogurt. Salt and pepper to taste.
Heat a grill or grill pan to high. Add the dried mint to the ground lamb and season with salt and pepper. Form into evenly sized patties. Apply some cooking spray to the grill. Add the patties, flipping after 4-6 minutes. Cook until lightly charred on the outside and medium on the inside. Remove from the grill and top with crumbled feta. Spread the greek yogurt sauce on the top bun and arrange 3-4 cucumber slices evenly.
Assemble the burger as follows:
Bottom bun, Burger with Feta, Pickled Red Onion, Baby Arugula, Top bun with tzatziki and cucumber.