Place spinach, Parmesan, walnuts, lime juice, salt, and garlic in the bowl of a food processor; process until finely chopped, scraping down sides of bowl as needed. With processor running, drizzle in oil, processing until well combined, about 30 seconds.
Cook spaghetti according to package directions, adding asparagus during the last 2 minutes. Drain and rinse under cold water. Place mixture in a bowl, and stir in 3 tablespoons pesto. Refrigerate until cold, about 30 minutes.
Divide noodle mixture among 4 shallow bowls; top evenly with tomatoes and remaining pesto. Serve immediately.