Preheat oven to 350°F. Line a 10x15-inch pan (jelly roll pan) with foil and spray with nonstick cooking spray (for easy removal).
Cream butter and sugar with a hand or a stand mixer until fluffy. Mix in egg and both extracts, then mix in baking soda, cream of tartar, and salt. Once mixed, add flour and stir carefully until cookie dough forms, then stir in chocolate chips.
Press mixture evenly in the prepared pan. Bake for 15-18 minutes, or JUST until it starts to get golden brown around the edges. These taste better if you err on the side of under done as they continue cooking as they cool. Cool completely before frosting.
To make the frosting: cream butter with a hand mixer in a large bowl until smooth. Slowly mix in powdered sugar until crumbly, then add salt, vanilla and 2 tablespoons of heavy whipping cream, adding more as needed for consistency.
Remove from pan and remove foil (if using). Frost sugar cookie slab with the frosting and sprinkle with crushed candy canes. Slice into sticks. Makes 64 cookies.
Store loosely covered for up to 3 days or freeze in a single layer for up to one month. Note: if freezing, add the peppermint crunch after defrosting or it will dissolve into the frosting.