The broken condition occurs at 50% hit point damage to an object (Usually. Some effects might give the condition without doing that much damage, but they will typically drop the item to half damage if they do.) Being broken isn't the same as being destroyed. An item doesn't typically lose its magical abilities from being broken or damaged. If destroyed, however, it will require make whole or similar effects to restore its powers.
A broken item (meaning an item with the broken condition) typically doesn't have to 'restore' its powers. If a living steel or greenwood magic items is brought to 0 hit points or 'destroyed' (which could occur from someone 'breaking it in half', not to be confused with making it 'broken'), it would lose its powers and also, since it's destroyed, wouldn't heal. (It is possible that someone may create a special material that could do such a thing, but greenwood and living steel both specify only being broken in the context of being damaged.)