As Mana mentioned, you'll get access to the Smoking Pipe at some point, which allows you to combat waves of enemies in return for fragments. You'll also eventually gain access to the Bedroll and Stockpot, which function in the same way, but offer variation on the narration and enemy types.
More specifically, the Smoking Pipe pits you against enemies from the Old World, the Bedroll pits you against Ura warriors, and the Stockpot pits you against beasts from the Wild. They all offer about the same reward as far as fragments are concerned, and without any bonuses to fragment collection you can expect to get approximately 750 fragments per run.
To increase this amount, consider enabling one or more idols in the Shrine, if you have one. Not only does this give a percentage bonus for fragments collected, completing these dungeons with at least one idol enabled completes a vigil at the Memorial, which will grant you an additional one-time lump bonus.