It can be argued that Amir is most like Father Ralph de Bricassart in The Thorn Birds and Jalil in A Thousand Splendid Suns. Let me explain.
All three men make poor choices in their lives, which lead to grief and tragedy. For example, in The Kite Runner, Amir does nothing while his childhood friend suffers a violent sexual assault at Assef's hands. Later, Amir's besmirching of Hassan's character leads to the latter being exiled (albeit reluctantly) from Amir's family home. Amir and Hassan become estranged for the rest of their natural lives. Filled with guilt and regret in later years, Amir tries to make amends for his actions by rescuing Sohrab, Hassan's son.
In A Thousand Splendid Suns, Jalil best corresponds to Amir. Although Jalil loves Mariam, he is reluctant to recognize her as his daughter during her childhood years. As a result, Mariam is forced to live with the shame of her illegitimacy. Like Amir, Jalil does nothing to protect those who put their trust in him. He (Jalil) allows his wives to decide Mariam's fate after Nana's suicide. Mariam is married off to the wicked Rasheed; she suffers grave physical and emotional abuse at Rasheed's hands for the duration of their marriage. In later years (much like Amir), Jalil tries to make amends for his cowardice. He leaves Mariam an inheritance (in a burlap sack) and writes an emotional letter to her. However, Mariam never gets to see the letter or to collect her inheritance. Like Amir with Hassan, Jalil never gets to reconcile with Mariam.
In The Thorn Birds, Father Ralph de Bricassart is most similar to Amir. Like Amir and Jalil, Father Ralph lives to regret his life choices. He chooses to pursue his ambitions in the Church rather than to recognize Meggie as his wife. Although he consummates his love for Meggie, Father Ralph is reluctant to abdicate his position in the papal hierarchy. He betrays Meggie's love and, as a result, never gets to know Dane as his son. Dane dies without knowing that the priest he admires is his father; ironically, it is Father Ralph who conducts Dane's funeral.
Hope this helps. Thanks for the question!