Preheat oven to 350 degrees F. Place whole mangoes in a shallow baking pan and roast until very soft, 70 to 90 minutes. Refrigerate until cool, about 1 hour.

Meanwhile, combine sugar and water in a small saucepan. Bring to a boil, stirring to dissolve sugar. Remove from heat and refrigerate until cold, about 1 hour.

When the mangoes are cool enough to handle, remove skin and coarsely chop pulp, discarding pit. Place the mango pulp and accumulated juices in a food processor. Add banana and lime juice; process until very smooth. Transfer to a large bowl and stir in the sugar syrup. Cover and refrigerate until cold, 40 minutes or overnight.

Freeze the mixture in an ice cream maker according to manufacturer's directions. (Alternatively, freeze the mixture in a shallow metal pan until solid, about 6 hours. Break into chunks and process in a food processor until smooth.) Serve immediately or transfer to a storage container and let harden in the freezer for 1 to 1 1/2 hours. Serve in chilled dishes.

MAKE AHEAD TIP: Store in an airtight container in the freezer for up to 1 week. Let soften in the refrigerator for 1/2 hour before serving.

Equipment: Ice cream maker or food processor