Wash and pat dry the tuna steaks. Season with salt and pepper and brush both sides with olive oil. Place tuna on hot grill and sear each side for 2 minutes for rate. If you prefer well done, cook the tuna an additional 2 to 3 minutes per side. Remove from skillet and let it rest.
Into the bowl of a food processor, add the garlic, basil leaves, pine nuts, salt, and pepper. Pulse until finely chopped. With the blender still running, slowly pour 1/2 cup of olive oil. Check for a thick, yet smooth consistency, adding more oil if necessary. Transfer to a bowl and stir in 1/2 cup Parmesan cheese.
Slice the tuna across the grain and on a bias into 1/2-inch thick slices. Place the slices on a serving plate and drizzle with lemon juice. Sauce tuna with pesto.