For the pickled onions: Combine the salt, sugar, vinegar, water, and hot sauce in a container with a lid. Shake to incorporate everything really well. Add the onions and let sit for at least 30 minutes.
For the burger: Preheat the grill. Combine all the burger ingredients and season well with salt. Form into 2 equal patties. Grill each burger for about 4 minutes on each side. During the last 2 minutes of grill time, top each burger with a slice of cheese.
Open the buns and place the inside on the grill, and grill until the bun is lightly toasted, about 1 minute.
To assemble: Schmear the inside of each toasted bun with the garlic mayo. Place the burger patty on the bottom bun. Top each burger with a few pickled onions, a tomato slice, and some arugula. Top with the top half of the bun.