Yes you can. I haven’t done it in years, but when I did, I just dipped the cutting in rooting hormone, stuck it in a pot with damp soil, and waited for it to grow. The cuttings rooted quickly with a high success rate.

what time of year woody???

I have rooted cuttings too dipped in a gel rooting hormone but it was in June. I would not expect much success at this time of year in our zones. Also curious to hear when you did yours woody.

It was a long time ago... but it was probably in late spring or summer when the plant was in active growth. The problem with rooting a cutting now would be getting it in the ground in time for it to get established before freeze-up. I’m not sure when that would be in Chicago.... But if whoever gave the OP the cutting might be able to give another one in spring if this one failed, I’d be inclined to give it a try and plant the rooted cutting either in a sheltered area close to the house to give it as much time as possible in unfrozen ground - or plant it in a large pot and store it in the garage for the winter. DH grows rosemary in a pot that he stores in the garage for the winter - rosemary would never survive outdoors in winter here but always survives in the garage. While I haven’t tried it, I’d bet lavender would do the same. Our garage is uninsulated and unheated. The pots are stored in the middle section of the garage. There are windows high up on the outer wall so there is light but it’s not particularly well-lit. What survives in there can be variable. The hardy begonia gdinie gave me last year didn’t survive and neither did a butterfly bush seedling. But rosemary did, as did garlic, and carrot and green onion seedlings. Leeks can be harvested during the winter from the pots in the garage. The OP probably has nothing to lose by trying... The alternative would be to try to grow it in a pot indoors. If the OP has a grow-light set-up it might work, but might not too....! Flip a coin! (Is it possible to take a cutting from the cutting and try it two different ways...?)

The cutting is pretty small. I was on a garden walk and didn't want to be too greedy, as I got other things as well.

I was actually thinking about trying it indoors while it is so small. I do not have grow lights, but I overwinter my rosemary indoors every year, in my brightest window.

Too late for this season though...and with lavender, I have found timing is far more crucial than many perennials. Taking fresh growth before flowering stems are formed in May/June is an ideal time...whereas cuttings taken after midsummer will tend to sit..., doing nothing.

Sounds like it could be 'Platinum Blonde'. If so, then be aware that it is a patented cultivar.

Rodney