For the onions: Preheat a grill pan over medium-high heat.
Drizzle the onion slices with the olive oil and sprinkle with the salt. Place in the hot grill pan and cook, 2 to 3 minutes per side. Remove to a bowl and toss with the balsamic and sugar. Cover the bowl with a piece of plastic wrap and let steam at room temperature for about 10 minutes.
For the avocado: Mash together the avocado, lemon juice, salt and pepper flakes in a medium bowl.
For the burgers: Heat the grill pan over medium-high heat. Form the beef into 6 even patties (about 1/3 pound each). Sprinkle the patties evenly on both sides with salt and place in the grill pan. Grill until nicely browned, about 2 minutes on the first side. Flip the burgers and cook for an additional 2 minutes. Place a slice of cheese on top of each burger and cover with a bowl or lid to allow the cheese to melt. Cook to desired doneness, about 1 minute longer for medium.
Place a small amount of arugula on the bottom half of each toasted bun. Top with the patties, tomatoes, if using, and the grilled balsamic onions. Spread some of the avocado on the top half of the buns and close the burgers.