Bring a large pot of salted water to a boil and cook the spaghetti according to package instructions until al dente. Reserve about 1 cup of pasta water before draining.
While the pasta cooks, add the fresh peas to the boiling pasta water for 1-2 minutes until bright green and tender. Immediately transfer them to a bowl of ice water to stop the cooking and preserve their color.
In a dry skillet over medium heat, toast the pine nuts for 2-3 minutes, stirring frequently, until golden and fragrant. Set aside to cool.
In a food processor, combine the basil leaves, toasted pine nuts, minced garlic, Parmesan cheese, and a pinch of salt and pepper. Pulse until finely chopped. With the processor running, slowly drizzle in the olive oil until smooth and creamy. Add reserved pasta water if needed to adjust consistency.
Drain the peas from the ice water and pat dry. Quickly sauté them in a little olive oil over medium heat for 1-2 minutes to warm through.
Return the drained spaghetti to the pot or large bowl. Add the sautéed peas and pesto, tossing gently to coat. Add reserved pasta water as needed to create a silky sauce.
Divide the pesto pea spaghetti among plates. Tear burrata into chunks and arrange over the pasta, allowing it to melt slightly.
Top with fresh basil leaves and extra Parmesan if desired. Season with salt and freshly cracked black pepper. Serve immediately.
