Heat olive oil in a large pot or Dutch oven.

Rinse and pat dry the chicken pieces.

Brown chicken lightly on all sides in the hot oil and remove to a plate.

Combine pomegranate molasses, ketchup, lemon juice, sugar, salt, and saffron threads in a bowl.

Sauté chopped onion in the same pot until golden.

Return chicken pieces to the pot with the onion.

Add the mixed sauce ingredients and chopped almonds.

Add enough water to just cover the chicken.

Stir and bring to a boil, then reduce heat to low.

Cover the pot loosely, allowing steam to escape.

Simmer for one hour, checking liquid level occasionally.

Add water if needed to prevent evaporation.

After 30 minutes of simmering, gently stir in the thick peach slices.

Continue to simmer until the chicken is cooked through and the sauce has thickened.

Serve hot over Iranian or Basmati rice.