Cut the fresh ube or yams into pieces to fit your pot. Fill your pot with water to completely cover all the pieces, then bring to a boil. Cook until a fork passes through the ube with little resistance. (about 1-2 hours)
Remove the yams and let them cool. While they are still warm, peel them and then mash them with a fork or potato masher. Personally, I use a blender
Add the sugar and condensed coconut milk and continue to blend
In a large non-stick pan or wok over medium heat, mix and stir until the halaya thickens considerably. Stir constantly or your halaya will burn. Don't be tempted to turn up the heat.
You'll know your ube is ready when your jam stays almost in place when you run your spoon through the middle of your pan.
Transfer to a heat-resistant container, let cool, then enjoy!
Notes
If you see large pieces of yam, mash them on the sides of your pan. But the blender should take care of that The halaya will continue to set and thicken as it cools and when refrigerated.