Add water to a medium pot and bring to a low boil. Add the apricots. Cook for 3 minutes. Remove from the heat, drain, and transfer to a bowl of ice water. Peel the skins from the apricots, remove the pit, and cut into fourths. Add the apricots to a food processor or blender and puree until smooth. Transfer to a bowl and set aside.

Combine the water and sugar in a medium pot and bring to a boil until the sugar dissolves. Stir in the puree, lemon juice, and honey. Remove from the heat and carefully transfer the mixture to a medium bowl. Cover and chill in the refrigerator until cool. Add the basil and stir until combined.

Transfer the bowl to a freezer and freeze the mixture for 3-4 hours. Remove the bowl from the freezer and gently mix with a fork. Return the bowl to the freezer.

When you are ready to serve, scoop into individual serving bowls or glasses with a spoon. Sprinkle with additional basil over the top, if desired. Serve immediately.