Broom should be cut when in flower, but before seeds are formed. It typically blooms in May. It is very important to cut at ground level, rather than pulling broom as seeds will germinate in disturbed soil. Cut plants do not need to be burned, the plant is not toxic and no seeds are present on the plant.
The plant may regrow from the root of well-established plants, but by re-cutting over a couple of seasons, the plant will weaken and die. The most important thing is not to disturb the root ball or the ground around the plant seeds in the ground are not stimulated to sprout.
Carol Swann