1. Preheat the oven to 425°F (220°C). Line a rimmed baking sheet with aluminum foil.

For the Soup

1. In the bowl of a food processor, combine the garlic and pine nuts. Process until finely chopped. Add the arugula, pecorino, lemon juice, and a pinch of salt. Process until coarsely chopped. With the processor running, slowly add in the olive oil in a steady stream. Process until smooth. Taste and adjust for seasoning with salt and pepper.
2. Place tomatoes on the baking sheet and drizzle with 1 tablespoon olive oil. Season with ½ teaspoon salt and ½ teaspoon pepper. Roast in the oven for 18-20 minutes, or until tomatoes are very juicy, but not broken down.
3. Meanwhile, cook the pasta in salted water until al dente according to package directions. Drain and transfer to a large bowl.
4. Add pesto, half the burrata, and basil to the bowl with pasta and toss until well coated.
5. Divide pasta between serving bowls and top with blistered tomatoes and remaining burrata. Garnish with additional arugula, pine nuts, and pecorino if desired. Serve immediately.