Line the 9x9 inch baking pan with parchment paper or grease it lightly with butter.
In a medium saucepan, combine the granulated sugar, cocoa powder, milk, and butter. Stir well to combine over medium heat.
Bring the mixture to a boil, stirring constantly. Allow it to boil for 5 minutes without stirring once it reaches a full boil.
Remove the saucepan from heat, and quickly add the peppermint extract and semi-sweet chocolate chips. Stir until the chocolate is fully melted and the mixture is smooth.
Fold in the crushed peppermint candies, reserving a few for topping if desired.
Pour the fudge mixture into the prepared baking pan and spread it evenly. If desired, sprinkle the reserved crushed peppermint candies on top.
Allow the fudge to cool at room temperature for about 2 hours.
Transfer the fudge to the refrigerator for at least 1 hour until set.
Once the fudge is firm, remove it from the pan and cut it into 12 equal pieces.