Pulled lamb: In a medium pot, combine all ingredients in the lamb mixture. Simmer on low until lamb is fully cooked and pulls apart. Remove lamb and let cool. Reduce sauce until it thickens to a barbecue sauce consistency. Remove excess fat from lamb and, using two forks, carefully pull lamb into thin strings.
Tzatziki Sauce: Whisk all sauce ingredients to combine; let sit to infuse flavours.
Assembly: Toast rolls or buns, if desired. Place pulled lamb on roll or bun, top with a spoonful of tzatziki.