Preheat your oven to 400°F (200°C). Spread the halved cherry tomatoes on a baking sheet and drizzle with 2 tablespoons of olive oil. Season generously with salt and freshly ground black pepper. Toss to coat evenly. Roast in the oven for 15-20 minutes, or until tomatoes are soft, slightly blistered, and caramelized around the edges. Remove and set aside.
Meanwhile, bring a large pot of salted water to a boil. Add the spaghetti and cook according to package instructions until al dente, usually around 8-10 minutes. Reserve about 1/2 cup of pasta water before draining. Drain the spaghetti and return to the pot or a large mixing bowl.
In a food processor, combine fresh basil leaves, grated Parmesan cheese, toasted pine nuts, minced garlic, lemon juice, and a pinch of salt. Pulse while drizzling in about 1/4 cup of olive oil until the mixture becomes a smooth, vibrant green pesto. Adjust seasoning with salt and pepper as needed.
Add the basil pesto to the drained spaghetti and toss gently to combine, adding reserved pasta water a tablespoon at a time if needed to loosen the sauce. Fold in the roasted cherry tomatoes carefully, so they retain their shape.
Divide the pesto-coated spaghetti and tomatoes between serving plates or bowls. Tear the burrata cheese into chunks and scatter on top of the pasta. The cheese will slowly melt into the warm pasta, adding creamy richness. Drizzle with a little extra olive oil and garnish with fresh basil leaves if desired. Serve immediately.
