So, I guess I need to know, do leeks send up additional plants from the base of the plant, or do they only produce by forming the flower bulb and seeds?

These are sending up lots and lots of side plants, which of course are very small but tall so they are basically clumping. And it tastes like an onion, not a garlic.

Thanks!

Yes, leeks sprout lots of pups around the older plant. They can be quite small and grass-like. Just dig them up and replant them with some room between and you are a go for leeks.

elephant garlic is a leek

Good point Michael

I have leeks, now into their second year. But I have not seen them
to grow shoots. Most prabably, as you alread mentioned that you
planted "Elephant Garlics" that is what you have.

Just a follow up here, have verification from several sources that these are indeed leeks, one from TAMU, and also from umext.umass.edu. Did more research on Leeks and found countless pictures of just how many pups these things can have!
I googled "leek how to propagate" and went the images at the top of the internet page, and wow, tons of pictures of leeks in all different stages.
Started seperating and thinning today, HUGE job.
I knew it wasn't elephant garlic as I have 4 rows of that planted on the other side of the garden, I have had that for almost 10 years now.
Thanks for all your help!
K

You are most certainly welcome!