Sicilian influence in New Orleans’ culinary canon is most prominently noted in the smooth, sweet tomato sauce known locally as “red gravy.” But immigrants from this Mediterranean island also introduced locals to a wide array of Italian sweet treats. Since 1905, the Brocato family has satisfied the city’s lust for frozen desserts to help beat the summer heat.

On steamy August nights, kids aged 1 to 92 line up outside the North Carrollton storefront as a uniformed sheriff maintains crowd con...