The good news is that this isn’t a branch, it’s merely a leaf stem. Losing a leaf shouldn’t be a problem for your plant. And yes, you can use it to propagate your begonia. In this case, I would recommend the following procedure
- Prepare a pot with moist, well-draining potting soil.
- Remove the leaf stem from the mother plant, cut the stem off the leaf, discard the stem.
- With a sharp and clean knife, cut/score the leaf veins from the underside of the leaf. The goal is to create small injuries that will trigger new growth. Scoring should be enough, if you cut all the way through the leaf it shouldn’t be a problem either.
- Immediately place the leaf on the potting soil (right way up, like it was on the plant) and anchor it to the soil. You can use U-shaped pieces of wire (think bent open paper clips), small stones or similar. The goal is to ensure that the leaf has a close contact with the soil underneath. Place the anchors between the slits, not on them.
- Cover the pot with a bit of transparent foil, slip it in a plastic bag or use a small greenhouse to ensure high humidity. Don’t forget to check the “greenhouse” every day and vent if necessary to prevent mold. Water carefully if needed.
- When little plantlets have formed, either gently cut through the leaf in between to separate them and give each it’s own pot or plant whole clusters of plants together for a fuller appearance.