1Combine sugar, brown sugar and butter in 2-quart saucepan. Cook over low heat, stirring occasionally, until butter is melted (6 to 8 minutes). Combine cooked mixture with ice cream in large mixer bowl; beat at medium speed, scraping bowl often, until smooth (1 to 2 minutes). Store refrigerated up to 2 weeks or frozen up to 1 month.
To serve, fill each mug with 1/4 cup mixture, 1 ounce rum or 1/4 teaspoon rum extract and 3/4 cup boiling water; sprinkle with nutmeg.