Using an electric mixer on low speed, blend softened butter, sugar, and salt until smooth and fluffy. Mix in the egg and vanilla extract until fully incorporated, scraping down the sides of the bowl as needed.
Beat in as much flour as possible with the mixer, then stir in any remaining flour by hand (you can also use the stir setting on a stand mixer).
Cover and chill for one hour. While the dough chills, preheat the oven to 375 degrees F. and line a large baking sheet with parchment paper.
Shape dough into one-inch balls; place 2 inches apart on the ungreased lined baking sheet.
Bake at 375 degrees for 8-10 minutes, until lightly golden around edges (very lightly golden…these should be soft when cooled.)
Remove from oven; lightly press a thumbprint in the center of each cookie (they don't have to be deep since you are piping icing on top). Allow cookies to cool completely.
While the cookies cool, make the icing: blend butter and extract. Gradually add powdered sugar and half & half to a piping consistency.
Pipe filling into centers with a star tip. Sprinkle immediately with crushed candy canes so they will stick before the icing starts to set.