You cannot pick up the Ender Dragon egg in vanilla Minecraft by hand
Break the block it is sitting on so the egg falls as an item
Use a torch, piston, or other block update method to make it drop
Collect the dropped egg item
On Bedrock Edition, the same general method applies: make it drop, then pick it up
If you want to move it without dropping it, use commands in Creative mode
If you are in Survival and the egg teleports, repeat the drop method until it lands where you want