a thing I figured you can do too is just tapping tiles around the dragon until you break the egg. I usually want to kill 5 HP worth of enemies, level up, 5 HP again, level up, then 6 HP and level up, then the next level requires you to get 9 XP with 6 HP. The egg gives exactly 3 XP so you can level up a 4th time without scrolls nor relying on chests