1. Fight the three bosses (the red doggo in the abyss dungeon, the rabbit hidden in the forest, and the kraken/whirlpool after defeating multiple octopus tentacles in the largest body of water outside the dungeon). Edit: I haven't fought or defeated the Sealed Pillar boss or found other bosses that drop the Philosopher's Fragment yet. The FAQ may have suggested there was only those three bosses in the current game/demo build.
2. Collect a Philosopher's Fragment from each of the defeated bosses
3. Use the escape mechanic, or use the Star tarot card item, to summon a red door and enter the door
Entering the door with a fragment in your inventory results in a cutscene that adds the elixir to your inventory. Then some combination of giving the sister milk to drink or returning home initiates the next cutscene to give the elixir to her (sorry I forgot).
Merlin's comment about their Philosopher's Fragment not turning into the elixir is likely that they repeated a boss fight, got the fragment, but mechanically that doesn't create a new elixir. Even though bosses do respawn, I think you only can create one elixir from each boss. Collecting extra fragments by repeating boss fights can be used as an item to increase your character's HP.
