You can definitely plant that cut off part. As already mentioned just stick in a cacti mix, make sure you use container with drain holes. I use big rocks around the base to help support it while it makes new roots. Being winter time be very, very patient. It is indeed a long wait. Do not water right away..wait a few days.
When I had a similar cutting I water far away from the base since there are no roots yet. Then next watering can take as long as it can. The big rocks supporting it helps me see if media is still wet. If it still shows damp delay watering some more. Typically it really slows down in growth during winter, so it is not that water needy yet and having no roots to take up moisture, the longer the delay to water is better for it
Root formation really takes time..can even go up to a month. It varies depending on the growing environment factoring ambient temperature, light levels and duration. So your patience will be your friend. You will know it is rooting once you see it grow at least one new leaf.or you can just see how it perks up better later on. Take photos so you can compare it after a month.
Make sure to position the plant in a warm, bright light area. No direct sun..it will eventually redirect all its energy to root formation.
