You can puree mangoes, as shown in the first step below, and freeze the puree to make drinks, pies, and ice cream all year long.
Combine the mangoes, lemon juice, and 2 tablespoons sugar in a nonreactive bowl; let sit for 2 hours. Puree in a food processor; strain to remove any fibers.
Using a mixer at high speed, beat egg and egg yolks until light and fluffy. Gradually add the remaining sugar, then lower the speed and add the milk, cream, and mango puree until well mixed. Chill, then pour into an ice-cream maker; freeze according to manufacturer's instructions.