Remove the steaks from the refrigerator at least 30 minutes before cooking.
Place the panini pan and press over medium to medium-high heat on separate burners until they are very hot. (The pan should not be smoking, but very hot).
Brush each side of steak with a few drops of oil. Season the steaks generously with salt and pepper.
Place the steaks into the hot grill pan. Using an oven mitt, carefully pick up the hot press and place it directly onto the steaks, covering completely.
Continue to cook for 4 - 7 minutes or to desired temperature (medium rare is about 125 - 130°F).
Remove the steaks from the grill pan and allow them to rest at least 5 minutes before serving.
While the meat is resting, cook the vegetables.
Place the vegetables on a tray, drizzle with the remaining oil and sprinkle with salt and pepper.
Place the vegetables into the hot grill pan and cook about 1 - 2 minutes, turning once.
Remove the vegetables from the grill pan and drizzle with balsamic vinegar. Serve.
(Note: Cooking times will vary depending upon the temperature of the steaks before grilling, the thickness of the steaks and the cooking temperature of the pan.)