Whisk together the flour, cocoa powder, baking soda, and salt in a bowl. Set aside.
In a large bowl using a mixer beat the butter and sugar on medium-high speed until smooth.
Add the egg and peppermint extract and beat on high speed until combined.
Mix in the dry ingredients on low until just combined.
Roll 1 inch sized balls of dough and roll each into sugar. Place them onto a lined baking sheet.
Using your thumb or a ¼ tsp sized measuring spoon, make an indent into the center of each cookie.
Chill the cookies for at least 1 hour in the refrigerator.
Preheat the oven to 350°F (180°C). Bake the cookies about 2 inches apart for 8 to 10 minutes or until the edges are set. Immediately out of the oven use the measuring spoon to reform the indent on each cookie. Allow to cool completely.
Make the chocolate ganache. Warm the cream in a small saucepan until it reaches a gentle simmer. Immediately remove it from the heat and pour it over the chocolate in a heat-safe bowl. Leave it for about 3 minutes, then stir until the mixture is thoroughly combined, and the chocolate has completely melted. Allow the chocolate to cool slightly before proceeding.
Spoon the chocolate into the indents of each cookie. Sprinkle with crushed candy cane.