There's no point doing anything until you are into the growing season, probably late spring. I'd take the offshoot off with a very sharp knife or saw and make sure you have plenty of root with it. Pot into a gritty compost and give some bottom heat if possible, and it should root well.
The only plants worth 'rooting' in the winter months are hard wood cuttings which root well during autumn and winter.

Thank you, that is exactly what I wanted to know. I just had a look at your website. What a lovely spot. I see that you have a gite. I also have a holiday cottage, next to my house here in Poole............small world.

Good luck with getting lots of visitors this year. I really enjoy showing our guests the gardens, though it's surprising how few people garden at all nowadays. We always give our guests a big basket of vegetables when they arrive, and you'd be surprised (or maybe not) how many just leave them and leave behind packet wrappings and cans from frozen and tinned food. Phillistines!
Never mind, all the more for us to enjoy!
Poole is a lovely spot, I always enjoyed visiting Brownsea Island and once taught lots of French students English there.

The bookings are quite good again this year, even thought they were a little later starting than usual. I now tend to give guests a welcome basket of teas, coffee , biscuits etc . As you say it is an amazing insight into peoples diet when you see the food that is left behind. Mind you we would think that as we are both veggies. I love Poole, always have and always will ! As it is rather mild here I can have a go at growing most things. Guests always seem rather impressed with the palm trees in the front garden.