Combine milk and vanilla beans in a saucepan; bring to a boil. Remove from heat and let cool to lukewarm, about 10 minutes.
Beat eggs and egg yolks together in a large bowl. Gently stir in confectioners' sugar, flour, and rum. Pour in warm milk mixture, discarding vanilla beans; gently stir until smooth. Cover bowl with plastic wrap and let batter rest in the refrigerator for 24 hours.
Coat fluted molds with butter and freeze until coating is set, about 10 minutes.
Preheat oven to 400 degrees F (200 degrees C).
Remove batter from the refrigerator, uncover, and pour into molds, filling each about 3/4 full.
Bake in the preheated oven until deep brown in color, about 50 minutes. Remove from oven and immediately invert each canele on a wire rack to cool, 30 minutes to 1 hour.