If you just want to rearrange your flowerbed, you should replant the whole 'artichoke' to a new place. Add some bonemeal to the soil. If you want to propagate your lily, you can try propagation by scaling. Good luck!

Normally most Lilies are of the "scaled" type as you call it. DO NOT break apart the entire bulb. You will lose the original one. You can remove a few flakes and plant those. 5-6 can be removed with no problem. If you get this done before winter the "new" lilies should grow a small stem and will most likely have one bloom the first year. They will bloom normally by the 3rd spring. The "artichoke' look is normal. You can also look for "new" bulbs growing arounds the stem just at the top of the bulb. About 1/4"-1/2" in size these will grow to full size in 3rd season also. There is a 3rd type which has very small"bulblets" growing on and attached to the stem next to each leaf. These will usually be on the upper half of the stalk. They are very small less than "bb" size. Wait until the leaves turn brown and remove/plant where you wish. Again 3rd season full size. Finally the seed pods can be propogated after they dry. Soon you have more than you want. Or more than anyone deserves like myself.