Cover the unpeeled ube with water and simmer for 30 to 40 minutes, until tender. Drain, cool enough to handle, peel, and finely grate or process until smooth.
Grease a shallow serving dish with 1 tablespoon butter.
Cook the Halaya:
Combine ube, coconut milk, condensed milk, evaporated milk, sugar, remaining butter, vanilla, and salt in a wide heavy skillet.
Cook over medium-low heat for 35 to 45 minutes, stirring and scraping constantly, until very thick and the mixture pulls cleanly from the pan in broad folds.
Set and Serve:
Spread into the greased dish and smooth with a buttered spatula, leaving natural swirls. Cool completely, then cover and chill for at least 2 hours.
Top with latik just before serving.
Notes
Frozen grated ube can replace fresh; thaw and drain it well before measuring.
Ube flavoring is optional and unnecessary when good fresh purple yam is available.