Beat the butter and confectioners' sugar together until just combined.
Mix in the vanilla, salt, and peppermint oil.
Add the flour and cocoa powder and mix just until a dough forms.
Fold in the chopped chocolates and crushed peppermint candy by hand.
Shape the dough into a log about 1 1/2 inches wide and 11 inches long, wrap, and chill at least 2 hours.
Let the log sit at room temperature 30 to 45 minutes to soften.
Slice into 1/2-inch rounds and place on a parchment-lined baking sheet.
Bake at 350F for 12 to 14 minutes until puffed and no longer wet-looking in the center.
Cool completely on the baking sheet before moving.