In a medium, heavy 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 to prevent it from forming a crust. Transfer the ube halaya to a container & set aside.
Ube Bread
In a small dish, heat milk to lukewarm. Add yeast & 1 tsp sugar; let sit for 5 minutes to allow yeast to activate. In a large bowl, whisk together remaining 1/4 cup sugar, melted butter, sour cream & egg. Add yeast mixture & stir to combine.
In another bowl, whisk flour & salt. Add flour mixture to yeast mixture 1 cup at a time, combining after each addition. Once all the flour has been added, knead on a lightly floured surface for about 2 minutes.
Lightly grease the large bowl, place dough in it & cover with plastic wrap & a tea towel. Allow to rest for at least one hour, in a draft-free place until dough has doubled in volume. Grate cheese & set aside in refrigerator until needed.
Grease 2 loaf pans. Punch dough down & divide in half. Roll each piece out to about 7×7-inch (18×18 cm) square. Spread ube halaya over each one leaving about 1/2-inch border all the way around. Sprinkle each square with Edam cheese then roll up in jelly-roll style. Pinch to seal seams & place seam side down in loaf pans. Cover with plastic wrap & a tea towel. Allow to rise for an hour or until doubled in size.
Preheat oven to 350 F. Bake bread for about 50 minutes until golden brown. Remove from oven; cool for just a few minutes then pat with butter. Sprinkle with more grated cheese.
Ube Bread Pudding (ON THE FOLLOWING DAY)
Preheat oven to 350 F. Butter a 8 x 8-inch baking dish. Cut thick slices of ube bread into 1-inch cubes & spread out in baking dish. In a saucepan, melt butter into milk on medium-high; do not let it boil. Remove from heat when butter is melted.
In a small bowl, whisk eggs & coconut extract together. Slowly pour egg mixture into butter/milk mixture & begin whisking immediately to avoid scrambling the egg mixture. Pour custard over the ube bread pieces; use a spoon to work liquid between the pieces.
Bake for about 30-45 minutes or until the top is springy. While the bread pudding is baking, make the ube drizzle.
Ube Drizzle
In a bowl, using a mixer, beat butter for 2 minutes then add cream cheese & beat for another minute. Add powdered sugar slowly then beat thoroughly; add vanilla. Mix in ube halaya until well combined. Add enough milk to make a drizzle consistency. Either drizzle bread pudding before or after cutting into serving pieces.
Recipe Notes
You will have extra Ube Bread & Ube Halaya but it will no doubt be eaten before you have a chance to freeze it. Other than just enjoying it by the slice, it makes great French toast as well.