If you like fresh pineapple, this ice cream is sure to become one of your favorites. For this recipe, the fresh pineapple should be very sweet; if it's not available, canned crushed pineapple is a better choice.
For a more adult version, add 2 tablespoons of dark rum during the final minutes of freezing.
Serve plain, with pound cake, or garnished with wedges of Roasted Pineapple. If time is short, mix the fresh chopped pineapple, juices and zest with 2 pints of softened, premium-quality vanilla bean ice cream to approximate the flavor.
Combine the cream, milk, sugar and salt in a medium-size saucepan, and bring to the start of a boil over medium heat.
Meanwhile, whisk the eggs in a medium-size bowl until combined. Very slowly add the hot cream to the eggs, whisking constantly. Return the custard to the saucepan and cook the mixture over low heat, stirring continually, until the custard thickens slightly and coats the back of a spoon, 4 to 5 minutes. If you have an instant-read thermometer, the temperature should be 180 degrees F.
Remove the custard from the heat and pour through a fine mesh sieve set over a clean bowl. Discard any solids. Stir in the lime zest and the lime and pineapple juices. Cool the custard at room temperature for 30 minutes, then place a round of parchment, or a piece of plastic wrap, directly against the surface of the custard, and refrigerate until the mixture is cold.
Freeze the mixture in an ice cream maker according to the manufacturer's directions. Add the chopped fresh pineapple during the last 5 minutes of freezing. If there's not enough room in your ice cream maker to do this, transfer the ice cream to a clean container and stir in the pineapple. Press a piece of paper or plastic wrap directly against the surface of the ice cream and freeze until hardened.