In a large saucepan over high heat, stir together mashed purple yam, sugar, coconut milk and sweetened condensed milk. Bring to a boil. Reduce heat to low and cook for another 15 minutes, stirring frequently. Pour into jar and set aside to cool.
Make the purple milkshake:
In a blender, combine purple yam jam, vanilla ice cream, ice and coconut milk. Serve with a dollop of purple yam jam, whipped cream, a sprinkle of purple yam powder and finish with a tuile cigar cookie.