1. In the bowl of a stand mixer or a medium bowl using a hand mixer, combine the butter with the granulated and brown sugars, vanilla, and water. Beat on medium-high speed until well-combined and fluffy, 3-5 minutes. 

2. Scrape the bowl down, reduce speed to low, and add the the flour, mixing until just combined and the dough is beginning to clump slightly. Mix in the chocolate and white chocolate peppermint chunks. 

3. Dump the dough out onto a hard surface and use your hands to bring it together into a ball. Divide the ball in two and form two logs, each 1 1/2 – 2 inches in diameter. 

4. Wrap each log tightly in plastic wrap and place in the refrigerator. Chill at least 3 hours, up to a couple of days (overnight is perfect if you have the time). 

5. When you’re ready to bake, preheat the oven to 350 degrees and line two rimmed baking sheets with parchment paper. Remove the dough logs from the fridge and allow them to soften slightly, 10-15 minutes. 

6. Unwrap one dough log and brush it all over with the beaten egg. Roll the log in the crushed peppermint candy, using your fingers if needed to press the coating into the dough. 

7. Using a very sharp knife, carefully cut the log into 1/2-inch slices. If the cookies crumble a bit, just mold them back together with your fingers. 

8. Place the cookies on the baking sheet, leaving just a little space in between them (once chilled, they won’t spread much). Return baking sheet to the fridge to chill slightly again, 10-15 minutes. 

9. Repeat steps 6-8 with the second log. 

10. Remove the first batch from the fridge and pop straight into a 350 degree oven. Bake for 13-15 minutes, until just barely golden on the bottom. 

11. Allow cookies to cool slightly on the baking sheet, then transfer to a baking rack to cool fully. 

12. Repeat with second batch. 

Cookies can be stored in an airtight container, in layers separated by parchment or wax paper, for up to a week.