You will receive Trade Quests  to 'Fix Broken [...] stone statue'. When you turn it in you will receive the sculpture.

Hime Stone Statues (a.k.a. Buff statues) can be created via a quest which requires 1 broken statue  This item is only dropped if you encounter and defeat a Hime as the last enemy before the boss fight. You must then carry on to defeat the boss to claim your prize.

Broken Stone Statue Items
Broken Kogarasumaru stone statue
Broken Komainu stone statue
Broken Akki
Broken Shirousagi Statue
Broken Tengu stone statue
Broken Tennyo stone statue
Broken Amen Statue
Broken Kabukihime Statue
Broken Otsuno stone statue
Broken Oyuki

For those that drop at Sculptor Mountain see:

For all others: