Brussels sprouts are a favorite in our house. Those tiny, delicious balls of cabbage will actually grow into a branch if left to themselves. Which makes them perfect for cuttings. Brussels sprouts take a long time to grow and mature, but starting with rooted cuttings speeds up the process considerably.
Brussels sprouts are a cool season crop. Once it gets too warm the sprouts will not form the tight little head. They just leaf out and grow into branches. You can leave a few sprouts on the plants for the purpose of propagating cuttings.
Wait until the new branch is at least 4 or 5 inches long, and is fairly thick and sturdy, them remove it at the base, next to the main stalk of the plant.
Remove all but the very top leaves from the cutting. Carefully nick or shave off a portion of the stem where the leaves were growing. Be careful not to remove too much of the stem. You need it to be strong and sturdy for planting.