Heat the oven to 300°F. In a small bowl, mix both sugars with the salt, cinnamon, and cayenne. In a large bowl, whisk the egg white until frothy; whisk in 1 Tbs. water until combined. Add the walnuts and stir to coat. Sprinkle on the sugar mixture and stir to evenly distribute.
Line a rimmed baking sheet with a nonstick baking mat or parchment. Spread the sugared nuts in a single layer on the sheet. Bake for 15 minutes,
stir the nuts, and continue baking until the nuts smell toasted and the sugar coating is caramelized, about another 15 minutes.
Let the nuts cool on the pan, separating them as they cool. When completely cool, transfer them to an airtight container.
Make Ahead Tips
Candied walnuts will keep for two weeks in an airtight container.