Combine 1 cup milk, sugar, and egg yolks in a small heavy saucepan; stir well with a whisk. Place over medium heat, stirring constantly. Cook for 8 minutes or until mixture reaches 180° and is slightly thickened (do not boil). Remove from heat. Pour custard into a cool bowl; place over ice water to cool completely. Combine 1/2 cup milk, dry milk, and vanilla. Stir in custard, banana, and syrup. Cover and chill.
Pour mixture into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions. Spoon ice cream into a freezer-safe container; cover and freeze 1 hour or until firm.