Hi Mary,

Thank you for submitting your question to the UVM Master Gardener Helpline.

Thistles can be challenging especially in an established bed where you need to be careful about disturbing the soil. Thistles grow well in open soil with lack of tillage, such as your asparagus bed. They have a deep taproot. In order to minimally disturb the asparagus, repeated hand removal (with good protective gloves, obviously due to the thorns) may be the most effective strategy in your asparagus bed.

In the field, however, repeated mowing can be effective especially after thistle flowering.  The roots are at their weakest right after the plant flowers. Continue mowing regularly to further weaken the roots. It can take two years or more to kill the plants this way.

Flaming is not an effective tool against thistles since it only burns the tops and does not impact the below ground rhizomes (spreading roots) that these plants use for reproduction.

The article below discusses control of thistles growing in asparagus beds 

https://blog-fruit-vegetable-ipm.extension.umn.edu/2020/06/weed-management-options-for-asparagus.html

If your thistle invasion is already widespread in your field, and repeated mowing does not provide sufficient eradication, you may want to contact a contractor to help. I have included a link to a list of contractors who offer assistance to landowners who want to manage invasives.
https://www.uvm.edu/sites/default/files/Extension-Community-Horticulture/Invasive_Plant_Control_Contractors_3-10-20201.pdf

Best of luck. I hope this is helpful. I battled thistles myself for years by hand-pulling before eradication.