Prepare the apple cider. Freeze the apple cider for a few hours until it reaches a slushy consistency.
Measure the oils and lye. Using a digital scale, measure out the lye, oils, and butter.
Mix the lye and cider. Carefully add the lye to the partially frozen cider and stir until completely blended. The mixture will melt, heat up, and may change color. Set aside in a safe place.
Melt oils and butter. Melt the oils and shea butter over low heat until completely melted and heated to 140°F (60°C). Set aside to cool.
Prepare mold and additional ingredients. Prepare your soap mold and any color or fragrance you plan to use.
Combine oils and cider mixture. When the oils and cider mixture cools to about 110°F (43°C), pour the mixture into the melted oils. Blend using a stick blender until "trace" is achieved – the soap thickens slightly, and a trail of soap rests on top of the batter rather than sinking in immediately.
Add fragrance. Stir in the fragrance oil by hand.
Pour into mold and smooth. Pour the soap batter into the prepared mold, gently smoothing out the top. Allow it to cure at room temperature.
Unmold and cure. Un-mold the soap after 24 hours, cut it into bars, and let the bars cure for 2 weeks before using.