In a large bowl, food processor or standing mixer, mix together the flour, sugar, baking powder and salt. Then add the egg, butter, rum, zest and enough milk, as needed, to form a smooth and pliable ball of dough.
Wrap the dough in plastic wrap (cling film) and refrigerate for at least 30 minutes.
On a lightly floured surface, roll the dough out into a thin sheet, not more than 5mm (1/4 inch). Cut the dough into festive shapes using cookie cutters. Gather up the scraps and form them back into a ball, then roll it out and cut it into more cookies. Repeat until you've used up all the dough.
Gingerly place the cookies onto a baking sheet lined with parchment paper. Brush the tops with egg white, then sprinkle them with the sprinkles.
Bake in a moderate oven (180C/350F) oven for about 12-15 minutes, until puffed up and cooked through. Take care not to brown them too much.