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 to prevent it from forming a crust. Transfer the ube paste/jam to a container & set aside.
Cheese
Grate cheese & set aside in refrigerator until needed.
Sweet Roll Dough
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 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.
Punch dough down. Divide into 18 equal pieces. Roll each piece with a rolling pin. Spread the middle of each piece with ube filling & SOME of the grated cheese. Close up the piece over the filling like an envelope, pinch long edges together, & roll it with your fingers into a rod shape. Coil each rod into a rounded snail shape. Place in greased baking dish & cover with plastic wrap/towel. Allow to rise for an hour or until doubled in size.
Preheat oven to 350 F. Bake rolls about 20 minutes or until golden. Remove from oven; cool for just a few minutes then pat with butter. Sprinkle with sugar & more grated cheese.