Heat half-and-half and brown sugar in heavy medium saucepan over medium heat to simmering.
Whisk about 1 cup hot half-and-half into egg yolks, then whisk yolks into remaining half-and-half. Cook, whisking constantly, over low heat until slightly thickened, 7-10 minutes. Strain through sieve and let cool to room temperature. Stir in rum and refrigerate covered.
Cut tops and bottoms from pineapples and cut away peels and eyes. Quarter lengthwise. Remove cores and cut each quarter lengthwise in half. Spoon sauce over pineapple and garnish with mint sprigs.