Chop basil, garlic and pine nuts in a food processor to a fine consistency.
Slowly add 1/4 cup of oil while blending to form a paste.
Add 3 Tbsp of Parmesan cheese, 1/2 tsp salt and 1/4 tsp of pepper and pulse until just combined.
Remove and allow the pesto to sit for 15 minutes.
Slice tomatoes in half and remove seeds.
Brush the cut sides with oil and season with salt and pepper.
Grill tomatoes cut-side down over medium heat for about 8 minutes. The tomatoes should be soft and have slight grill marks.
Carefully remove from the grill and stuff each half with the pesto blend.
Sprinkle Parmesan cheese over each and serve.