Preheat the oven to 350°F. Spread the nuts on a parchment-lined baking sheet and toast until fragrant, about 8 minutes. Let them cool completely. (Leave the oven on.)

In a medium bowl, whisk the egg white and 1 teaspoon of water until very frothy. Add the granulated sugar, brown sugar, pumpkin pie spice, salt, maple or vanilla extract, and cayenne pepper. Add the nuts and toss to coat evenly. 

With a slotted spoon, transfer the nuts back to the parchment-lined baking sheet, discarding any extra egg white mixture in the bowl. Roast until the nuts are toasted and fragrant and almost completely dry to the touch, 15 to 20 minutes. (The nuts will continue to crisp as they cool.) Once the nuts are cool, break up any large clumps. Transfer to an airtight container and store at room temperature for up to two weeks.