Chocolate Mint Thins with Candy Cane Crunch

Ingredients

• 1 cup plus 2 tablespoons granulated sugar
• 12 tablespoons unsalted butter, softened at room temperature
• ½ teaspoon kosher salt
• 1 egg
• 1 teaspoon peppermint extract
• 1 tablespoon water
• 1½ cups all-purpose flour
• ¾ cup unsweetened cocoa (preferably Dutch-process)
• 8 to 12 candy canes
• 16 ounces semisweet or bittersweet chocolate
• ¼ teaspoon vanilla extract
• ¼ to ½ teaspoon peppermint extract
• 2 tablespoons coconut oil