Line a strainer with four layers of cheesecloth or one coffee filter and place over a bowl. Place yogurt in prepared strainer; cover yogurt with edges of cheesecloth. Refrigerate for 8 hours or overnight. Discard liquid from bowl. Place yogurt in a large bowl; stir in the basil, sugar, walnuts and lemon zest.
Fill cylinder of ice cream freezer; freeze according to the manufacturer’s directions. Transfer to a freezer container; freeze for 2-4 hours before serving.