Combine oil, basil, salt and pepper in a blender and blend for 1 minute. Strain into a bowl, discard solids
Combine tomatoes and onion in a bowl, add a few tablespoons of the basil oil and season with salt and pepper. Reserve remaining basil oil for Shrimp recipe.
Heat grill to medium heat. Brush avocado with oil on grill and season with salt and pepper. Grill cut side down until lightly golden brown, about 2 minutes. Remove to a platter, cut side up. Place some of the tomato mixture on top of each avocado, drizzle with more of the oil and sprinkle with the pine nuts and cheese. Garnish with basil leaves.