Cream butter & sugar in the bowl of a stand mixer on high speed until light & fluffy. Add the egg and vanilla and beat until well combined.
Split dough in half and shape each half into a disc. Wrap in plastic wrap and chill in the freezer for 1 hour, fridge for 2 hours. At this point, you can also put the dough in the freezer to use at a later date.
Remove the cookie dough discs from the fridge or freezer and unwrap. If using frozen dough, allow to thaw in the fridge for an hour or so.
Roll the dough out until it is about 3-4mm thick. If the dough is too hard to roll out, leave it sitting for 10 minutes or so and then try again. Using a gingerbread man cutter, cut out the cookies and lay them on a lined baking tray. Re-roll any scraps of dough to make more cookies. Bake for 12-14 minutes, checking them at 12 minutes, until they are starting to harden and are looking slightly darker around the edges.
Melt the white chocolate buttons* in a microwave safe bowl for 20 second bursts, stirring after each time in the microwave, until completely melted.
Pour the white chocolate into a small snaplock bag, seal it and make a small cut in the corner of the bag to pipe out of.
Use a small amount of white chocolate to stick the candy eyeballs onto the mummy cookies. Gently squeeze the snaplock bag to decorate the cookies. If the chocolate starts to harden, run the bag under warm water until it softens again.
Allow the chocolate to harden before serving.
Store cookies in an airtight container for about 1 week.
* The chocolate will harden eventually so it might help to melt the chocolate and decorate the cookies in 2 batches. And I would recommend just throwing out the bag once you're done rather than trying to refill it!