In a medium saucepan, melt butter. Add coconut & condensed milks; stir until heated. Add thawed, grated ube & stir everything together. This process takes about 40-50 minutes until the ube is cooked. The mixture will be thick & sticky. It is important to stir the mixture often during cooking process to prevent it from forming a crust. When cooked, transfer the ube halaya to a container. Cover it with plastic wrap, making sure it is touching the ube paste to prevent a skin from forming & set aside. I find, making this filling a day ahead makes everything come together a lot easier.