Answer: I'd pull the thyme out, because herbs can become quite invasive and you don't want anything competing for nutrients and water with your asparagus.
When harvesting asparagus, if you use a knife, you may injure the crown, but snapping the spears off will keep the crowns safe from injury. Let those forgotten spears grow into fronds to help recharge the roots for next year's harvest. (A few of those fronds look great in floral bouquets as greenery!)
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