[Tilted Irish Kilt]

painter :" We cleaned out the weeds and composted them.
For the last couple weeks we have been cutting asparagus like crazy."

I have heard to leave the skinny asparagus shoots in place to build up strength and production for next year.
The foliage that they produce provide the nutrients and photosynthesis to improve vigor for next year, just as you dont harvest a new bed for 2-3 years.
However, I will defer to Augie, since he is the commercial grower, and has more practical experience than me .

[Augie]

I have heard to leave the skinny asparagus shoots in place to build up strength and production for next year.

Those are on there because I wasn't out there picking when I should have been picking. lol

I planted my asparagus bed in 2008 and didn't pick anything from it the first three years. Very light pickings the next two years. Last year I picked it clean til the end of May and I plan on working it the same way this spring. I add 2"-3" of compost on top every year after the tops die back. It seems to be working.

As far as letting it go to seed... male plants will give you more asparagus to eat. Female plants will give you wild asparagus all over the neighborhood. I leave one female plant in my patch just for that purpose. Any new ones that show up get plucked and chucked.