Steam or boil the peeled and grated ube until very tender. This usually takes about 20-30 minutes. Once tender, mash the ube thoroughly using a fork, potato masher, or food processor until smooth. In a large, heavy-bottomed pan or wok, melt the butter over medium heat. Add the mashed ube, condensed milk, evaporated milk, and granulated sugar. Stir well to combine. Continue to cook the mixture over low to medium heat, stirring constantly to prevent sticking and burning. This process is crucial and can take 40-60 minutes, or even longer, until the halaya becomes very thick and pulls away from the sides of the pan. If using, add the ube extract during the last 10-15 minutes of cooking and mix thoroughly. Once the ube halaya reaches your desired consistency, transfer it to a lightly greased pan or mold. Flatten the top with a spoon or spatula. Let it cool completely to room temperature, then chill in the refrigerator for at least 2-3 hours, or until firm. Serve cold, optionally garnished with grated cheese or toasted coconut flakes.