First make the Thai basil simple syrup. Combine the sugar and water in a small saucepan on the stovetop and bring them to a boil. When the syrup reaches between 215-220°F (101.5-104.5°C) turn off the burner and add in the basil. Set aside and let the syrup cool to room temperature.
Next prepare the other ingredients for the blender. For the pineapple, cut off the top and bottom to create a flat cutting surface. Then, use a chef knife to cut down the standing pineapple and split it in two. Take one half of the pineapple and split it into quarter and then eighth wedges. For each wedge, slide the knife between the flesh and the outer skin and then separate the flesh from the inner core to form a usable plank of fruit. Roughly chop the pieces and add them to the blender until the correct amount has been chopped.
For the mango, use a paring knife to cut from top to bottom as closely to the pit as possible on the widest sides. Then, use a spoon to scoop the flesh of the fruit from the peel. There may be some smaller strips of fruit you can take off from the narrow sides and bottom as well. Rough chop these pieces and add them to the blender.
Next, zest the limes with a Microplane or zester. Slice them in half and squeeze the lime juice. Then, use the edge of a spoon to scrape the skin off of your fresh ginger. Grate the ginger until you have enough for the sorbet. Measure out the salt, Cointreau, and Thai basil leaves. Add everything to the blender.
By now, the Thai basil syrup should be cooled. Strain the simple syrup and pour that into the blender. Turn your blender to high for 1-2 minutes to make sure everything is well broken down.
Pour the blended ingredients into a medium sized bowl and refrigerate until very chilled. At least 1 hour. After chilling, pour the final mixture into an ice cream maker. Freeze to the manufacturer's instructions and when done, scrape out of the mixer and into a freezer safe container. (see notes: if you don't have an ice cream maker)
Freeze for at least another 4 hours or overnight for best results.