In a small heavy bottomed saucepan over medium heat, add the sugar, water or pineapple juice, lemon juice and zest.
Stir just until sugar melts; let cool.
Transfer pineapple mixture to a food processor.
Add 1 cup buttermilk; puree until smooth.
With the motor running, slowly add remaining buttermilk just to blend - don't overfill.
Taste to see if it is sweet enough – adjust as needed.
Pour into a freezer-safe dish; place in the freezer.
Freeze until firm, at least 2 hours.
Notes
* Note I used an ice cream freezer to freeze. Be sure your canister is frozen the day before. Or use a metal loaf pan or a flat glass dish that is freezer-proof. To store, place a piece of plastic wrap and then some wax paper over the top, then a lid; it seems to attract the ice crystals and not on the sherbet. Top with toasted coconut and toasted macadamia nuts, or eat plain.
Recipe by Make Mine Lemon at https://www.makeminelemon.com/pineapple-buttermilk-sherbet/