In an electric stand mixer, add butter and both sugars. Blend until they are light and fluffy.
Add caramel extract and both eggs.
Add flour, baking soda, and salt. The dough should no longer be glossy, but if it is still too sticky add 1 tablespoon of flour at a time until it is the right consistency.
Chill the dough for 30 minutes in the refrigerator.
When ready to bake, preheat the oven to 375℉. Line a baking sheet with parchment paper or lightly spray with nonstick cooking spray.
On a lightly floured surface, roll out dough to ¼" thick and use cookie cutters to make shapes. Place the cookies on the baking sheet.
Bake in batches in the oven for 8-9 minutes until the tops spring back when lightly touched. Place them on a cooling rack and bring to room temperature before icing.
Heat bittersweet chocolate squares in a small bowl in the microwave at 30-second intervals until melted. Let cool slightly and add to an icing bag. Add caramel syrup to another icing bag.
Notes
Caramel extract can be tricky to find. You can sub for vanilla extract but you'll have a milder caramel flavor in the cookie itself. Adding the caramel syrup can bring more caramel flavor back.
I recommend using bittersweet baking squares to melt down to a liquid for finer drizzling. Bittersweet chocolate chips have a stabilizer in them that prevents them from becoming a liquid. They do melt, but it is thicker and does not drizzle as finely.