Make orange ice: In a large bowl, combine orange juice, lemon juice, liqueur, and 1 tablespoon sugar. Stir until sugar is dissolved. Pour mixture into a 9-by-13-inch baking dish or pan. Cover with plastic wrap; freeze until firm, about 3 hours.
Using a vegetable peeler, remove zest from 1 orange (leave white pith behind); slice zest into long thin strips, and set aside. Set aside orange.
Make candied zest: Heat 1/2 cup water and remaining 1/2 cup sugar in a medium saucepan over medium-high heat, stirring until sugar dissolves. Add reserved zest; bring to a simmer. Cook 2 minutes. Using a slotted spoon, transfer zest to a bowl; set aside. Reduce heat to medium.
Make compote: Scrape seeds from vanilla bean into pan; add vanilla bean and cinnamon stick. Bring to a simmer; add cranberries. Cook until berries burst and mixture thickens, 6 to 8 minutes. Remove from heat, and let cool. Discard vanilla bean and cinnamon stick.
Peel remaining 2 oranges with a paring knife, following curve of fruit; remove pith from reserved orange. Working over a bowl to catch juices, cut out segments from membranes. Squeeze membranes over bowl; coarsely chop segments. Add oranges and juice to cranberry mixture. Transfer to a bowl; cover with plastic wrap. Chill at least 1 hour and up to 2 days.
Using a fork, scrape orange ice to create a slush. Spoon into serving dishes; top with compote and candied zest.