Arrange the swordfish steaks in a nonreactive baking dish. Season on both sides with salt and pepper. Squeeze the juice from the remaining lemon over them. Sprinkle with oregano and drizzle with olive oil. Turn the steaks a few times to coat with the flavorings and marinate in the refrigerator for 30 minutes.

Set chimichurri aside.

Meanwhile, build a wood fire in your grill with a hot zone for searing and a medium zone for cooking. Brush and oil the grill grate.

Arrange the swordfish steaks over the hot fire and grill until the fish is browned and sizzling and cooked through, 4 to 6 minutes per side, 8 to 12 minutes in all. Move the steaks to the medium zone if they start to burn. Baste with a little of the chimichurri (pour it in a separate bowl) as the fish cooks.

Serve the swordfish steaks with the remaining chimichurri on the side for spooning over the fish.