Heat a grill pan over medium-high heat. Coat with grape seed oil.
Form patties of ground beef using a ring mold. Season both sides heavily with salt, and the top with pepper. Put the patty on the grill, pepper side down, and season the top with pepper. Cook for 5 minutes on each side. While the burgers are finishing, add two lemon slices to the grill pan to char on either side.
Put lemon juice, olive oil, Dijon, and a pinch of salt in a Mason jar. Cover and shake.
Lightly dress the arugula in the lemon olive oil dressing.
To assemble, place burger on buns, topping with charred lemon, dressed arugula, and sliced cheese. Drizzle with balsamic reduction and serve.