Place yams into a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain and cool until safe to handle, about 20 minutes. Grate yams into the pot.

Add coconut milk, sweetened condensed milk, brown sugar, evaporated milk, butter, and vanilla extract to the pot over low heat. Cook, stirring constantly, until thick and sticky, about 20 minutes; mixture should be able to drop into room-temperature water and form a small ball when done.