At grocery stores in our area they prepare cactus pads for sale in-house. I haven't sat and watched them for hours but it seems that the process involves a pair of gloves and a knife. The gloves are generally the cloth gloves with rubberized palms/fingers and the knife is a chef's knife.
This image shows the process I usually see them following, though they're using a much larger knife in the store... admittedly, they do it on a daily basis, so it might be good to start with a smaller knife:
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This image is from what looks like a nice guide on preparing cactus pads.
The steps are pretty simple:
VERY IMPORTANT TIP: Use kitchen tongs to hold the nopales or, better yet, wear gloves to protect your hands from the thorns. Once you are familiar working with cactus paddles then you can try to clean them without gloves.
- Place the cactus paddle on your cutting board and, using a sharp knife, trim off the edge. Scrape the spines, thorns or eyes, running your knife from back to front until completely clean. Turn the cactus paddles and do the same on the other side.
Other sites seem to echo this process, so it seems pretty universal. One site even recommends replacing the knife with a spoon, which is less likely to dig into the cactus pad, particularly if it's convex.