Definitely, I would plant them in the garden and let them over winter.

Are you certain they're leeks? Leeks don't bulb and the one on the right looks as if it has done so. You also say you discarded the bulb when you cooked with them. Leeks shouldn't have a bulb to discard. You eat the whole thing bar the basal plate with the roots on. The one on the left looks as if it is putting up a flower stalk. Could they be elephant garlic? Anyhow, there is nothing to do with them except put them in the ground and see what happens.

Flora, you got me thinking:
There is nothing on the garlic to be discarded. But you can root the bottom few inches of leek, just like green scallions.

Another possibility is that OP (a shau) used a young/green garlic (?!?)
let us see what kind of explanation "a shau" has.

The one in the pot (IF indeed it is a leek--as a previous poster pointed out the one in the water sure looks like garlic) will never develop into the size a leek grown in open ground would be. I plant my seedling leeks in the spring in a trench at least 6-8" deep and gradually fill in the soil along the trench as the plants grow. This gives me a nice long white part in the fall, much longer than one usually sees on grocery store leeks.

I must add that if the allium in the water is garlic, I cannot believe a bulb would from a piece of root in water. Garlic cloves must be planted in fall, overwintered, and the full bulb is not ready for harvest until the following July.

Are you pulling our collective legs here???

Does the plant in water, on the right has a bulb?
The answer is: It is not certain. What we see as a BULB could be an optical illusions

When you try to root a bottom part of a leek, Only certain leaves from the center will grow. That is why you will get a leek with a much narrower stem than what it originated from . I experimented rooting from leek's bottom years ago. Even split it into pieces to see if can get more than one plant... anyway, since garlic does NOT have a throw-away part, and leeks have the same shape of leaves, make the matter confusing. I am sure OP can clarify all of this guessing.

They have flat leaves so they can't be scallions. They have to be either leeks or elephant garlic. I'm going to say that they are leeks and that the "bulb" we see is actually the leftover part of the original leek.

Yes, you should plant them outside as soon as possible.

Rodney

Hey guys - thanks for all your responses. These are leeks. I probably just used the wrong terminology. What I meant was after cutting off the ends with the roots and eating the top white & green parts, I planted that part with the roots in water, and within a day they started shooting up.

The new leaves are in no way as fat as the original plant, whose original leaves were probably an inch wide. But yes, these are in fact leeks.

OK, so if they are leeks they are essentially biennial and will flower in their second year from seed. So the plants you have are going to flower either this year or next. I'm not sure what your goal is with rooting them. Is it curiosity to see what happens or do you hope for another harvest from them? If you want good quality leeks for eating it would be best to start again from seed. You don't say where you live so we can't say if these will over-Winter in your zone. Personally, I think the one on the left is going to flower soon anyway.

hey all - again, thanks for your responses! yes my object was really just curiosity and now i have seen that they grow really well. maybe not for eating, but just as plants. i live in SF, Ca in a transitional/partly foggy area. I guess I'll plant them in the yard and see what happens. Thanks!