Beat sugar and cream cheese together until smooth and creamy. Beat in egg and vanilla; set aside. Quarter some strawberries and mash the others. Combine strawberries and brown sugar in small bowl; set aside. Bring half-and-half to a boil in heavy, medium saucepan. Slowly beat the hot cream into the cheese mixture.
Pour all back into pan; place over low heat. Stir constantly with whisk or wooden spoon until custard thickens slightly. Be careful not to boil or the mixture will scramble. Remove from heat; pour though strainer into large, clean bowl. Allow custard to cool slightly, then stir in orange zest and cream. Cover and refrigerate overnight or until cold. Stir chilled custard; freeze in 1 or 2 batches in ice cream machine according to manufacturer's instructions. Add strawberries when ice cream is semifrozen and allow machine to mix in, then stir in the jam or jelly and do not overstir. When finished, ice cream will be soft but ready to eat. For firmer ice cream, transfer to freezer-safe container and freeze at least 2 hours.