Make flax egg: Mix ground flaxseed and water in small bowl and place in the fridge for 5-10 minutes to set.
Using a hand mixer, beat butter and sugar until fluffy.
Add flax egg, vanilla, applesauce, apple cider vinegar, and molasses to wet mixture and beat until just combined.
In a separate bowl, whisk together dry ingredients. Add the dry mixture to the wet ingredients and beat until just combined.
Wrap the dough in plastic wrap and place in the fridge for 30 minutes to chill.
Once the dough is chilled, roll out to 1/4-inch thickness between two sheets of wax paper. If the dough is sticking to either side of wax paper, sprinkle some flour on the surface of the dough.
Use cookie cutters to cut out shapes. Continue re-rolling out the dough to make shapes.
Transfer cookies to a baking sheet lined with parchment paper or a baking mat. Pop the baking sheet into the freezer.
Preheat the oven to 350 degrees Fahrenheit.
Remove cookies from freezer and bake for 8-10 minutes.
Remove cookies from oven and let cool completely before transferring to cooling rack.
To Make Icing:
Whisk all ingredients for icing in a small bowl. Transfer to a piping bag or plastic bag and cut a small corner of the bag. Decorate cookies as desired!
Video
Notes
Instead of waiting for the dough to chill, you can make the day before and refrigerate overnight.