I'd say cut and eat them until they come up as thin spears. Then let them grow as plants for maybe 6 months or so. If you have a sort of winter - cooler weather - then cut the plants back, fertilize, and wait for the big ones to come up again.

Do you have any kind of Agricultural School or Department that you could ask?

It never really gets that cold in Southern California so this link might help: http://mgorange.ucanr.edu/Edible_Plants/?ds=530&uid=154

Thanks Kate! It looks like in this "first" harvest, we already have probably half as very thin spears. As such, we'll harvest the thick ones and let the thin grows grow for a few more months.

There is probably some sort of Agricultural School, but honestly, I'd have to do some digging to find contacts. But more so, this isn't a super common vegetable here in Colombia. They are starting to grow it, but a lot is actually still imported.

Thanks for the link, I'll check it out.

When I was reading about asparagus, I think I read that it needs a dormant season. This is normally a cold winter (below freezing), but huge quantities are successfully grown in California areas where the weather does not get cold enough, so they give it a dormant season by withholding water.