Add the yeast and a spoonful of sugar to slightly warm milk. Allow to sit for about 10 minutes.
In another medium bowl, mix the rest of the sugar with the egg and egg yolks, lemon zest, butter, vanilla, rum, and salt.
Now mix the yeast mixture with the egg mixture. Knead the dough. Cover and leave for 2-3 hours. It should be about double in size when ready. Time will depend on how warm it is in the room.
Turn the dough out on a lightly floured area. Flaten dough until it is about 1/4 inch thick. Then use a cutter to cut the dough in about 2-inch circles. Allow cutouts to rest for about 20 minutes.
Heat oil on medium heat.
When the oil is hot, cook each dough cutout for 1-2 minutes on each side. Remove from oil and let cool on a rack.
Mix all the cheesecake filling ingredients together until smooth.
Add the cheesecake mixture to a piping bag and push the end in the side of a doughnut and squeeze the bag to stuff the donuts. Pour or dip the icing over the top of the stuffed donuts. Then sprinkle with graham crumbs over the icing.