In a wide, heavy-bottomed pan over medium heat, combine grated ube, coconut milk, condensed milk, evaporated milk, condensed milk, butter, and sugar.
Bring to a boil, stirring occasionally until sugar is dissolved, butter is melted, and ube is dispersed.
Add ube extract, if using, and stir until mixture is evenly colored.
Lower heat and continue to cook, stirring regularly, for about 30 to 40 minutes or until a soft, sticky dough forms.
Add cheese and stir until melted.
Continue to cook for another 10 to 15 minutes or until mixture is thick enough to cling to the back of the spoon and pulls slightly from the side of the pan.
Remove from heat and divide equally into the four prepared llaneras. Use an offset spatula to press and flatten the ube evenly.