Preheat the oven to 350 degrees F. Grease a large sheet pan with butter. Place the oranges in a saucepan over medium heat and cover with water. Bring the oranges to a boil and cook until soft, about 20 minutes. Remove the oranges and cool. Using a knife, remove the zest and fruit, being careful not to get too much of the white pith with the zest. In a food processor fitted with a metal blade puree the zest and fruit until smooth, you should have 1 1/2 cups of orange puree.
Combine the butter and the sugar in the bowl of an electric mixer and mix until fluffy, about 3 minutes. Add the eggs and cointreau and mix until incorporated. Add the pureed orange mixture and mix until incorporated. In a separate bowl, combine the dry ingredients. Add the dry ingredients to the batter and mix until well incorporated.
Pour into prepared pan and bake until the center springs back, about 25 to 30 minutes. Remove from the oven and cool. Cut the cake into 24 equal pieces. To assemble, place one piece of the cake in the center of 12 plates. Place 1/4 cup of the ice cream on top of each piece of the cake. Spoon some of the compote over the ice cream. Place another piece of cake on top of the compote, forming a sandwich. Repeat the above process with the remaining ice cream and compote. Garnish with mint sprigs and a sprinkle of confectioners' sugar.
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