Cook pasta according to directions and then cool under cold running water, lay out on a towel to dry while you prep the rest.
In a large bowl combine the pesto, lemon juice and olive oil. Whisk together and then add pasta and asparagus. Combine gently until the vinaigrette coats everything.
Add asparagus and toss together gently.
Lay salad out of a platter and top with bit of torn burrata (or keep it whole and take a chunk with each serving), season with salt and pepper and drizzle with a little olive oil. Serve immediately.
To make this ahead, do step one and two adding the asparagus but stopping before adding the arugula. Store in the fridge for a few hours, adding the arugula and burrata right before serving.