Slice each layer of 12 rolls horizontally, keeping them attached to each other. Set aside.
In a stand mixer fitted with the paddle attachment, beat the cream cheese, sugar and vanilla until soft, 3 to 4 minutes. Scrape down the sides of the bowl and the paddle with a rubber spatula. Add the cream and beat for 1 minute.
Lay out half the rolls, the layer with the tops on your left side and the layer with the bottoms of the rolls on your right. Spread half of the cream cheese mixture over the cut side of the tops, spreading to the edges. Spread about 1/3 cup of the Nutella over the cut side of the bottom half. Sprinkle the Nutella with half of the crushed cookies. Top with half the cherries, placing 3 to 4 cherry halves on each roll.
Carefully flip the top of the rolls, cream cheese side down, over the bottom of the rolls. Repeat with the remaining slab of 12 rolls, the cream cheese mixture, Nutella, crushed cookies and cherries. (These can be made to this point a day ahead and wrapped and refrigerated.)
Heat a panini press to 350 degrees F. Spread the soft butter on the top of the rolls and sprinkle with the turbinado sugar. Put one slab of the panini on the hot press and close the lid, taking care not to press too hard (don't squeeze all the goodies out of your panini!). Cook until golden brown and caramelized. Remove from the press and repeat with the other rolls. (Once caramelized, the paninis can sit for a few hours. To reheat them, pop them into a 300-degree F oven until warm, 6 to 8 minutes.)
Separate the panini into mini rolls, carefully cutting each slab into 12 pieces. Dust with confectioners' sugar and serve.