Wash the persimmons, dab dry and cut in half horizontally. Cut the round sides straight so that they do not tip over. Place the persimmons in an ovenproof dish.
Preheat oven (electric cooker: 200 °C/circulating air: 175 °C/gas: level 3). Coarsely crumble the Amarettini. Mix with rum and sugar and spread on the persimmons. Bake in a hot oven for 10-12 minutes.
Meanwhile, wash the lime in hot water and grate it dry. Rub the peel off half the lime. Squeeze the lime. Mix yoghurt, honey, lime zest and juice. Sprinkle persimmons with sugar crystals and serve with lime yoghurt.