In a bowl gently combine beef and thyme and form into two 3/4-inch-thick patties. In a heavy skillet (preferably cast iron) heat oil over moderately high heat until hot but not smoking and reduce heat to moderate. Cook patties 3 1/2 minutes on each side for medium-rare meat and drain on paper towels.
Spread mayonnaise on insides of rolls and buns and make sandwiches with onion, arugula, and patties.
