Preheat grill to 400 degrees F.
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 rare. 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.
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.
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