In a large bowl or with your stand mixer cream the granulated sugar and butter until it is fluffy. If using the stand mixer it will take about 5 minutes.
Once the sugar and butter is creamed together, add in the egg and vanilla extract.
In a medium sized bowl mix the flour, baking powder, salt. Use a whisk to mix everything together thoroughly.
Slowly add the flour mixture to the sugar, butter, egg and vanilla extract mixture. Mix everything until well incorporated.
Put a handful of the freshly made dough between two sheets of parchment paper and roll until you get your desired thickness and set aside still on the parchment paper.
Repeat with the rest of the sugar cookie dough.
Put the rolled dough including the parchment paper, on a cookie sheet and refrigerate for at least 10 minutes. You can keep it in the fridge for longer if needed.
Once your dough is chilled completely it is ready to use cookie cutters to make the cookie shapes you are wanting to make.
Use your carrot shaped cookie cutters and cut out your cookies. Then reroll the scraps and continue cutting out cookies until all of the sugar cookie dough is used up.
Place cookies on a baking sheet and bake at 325° F for 10 -15 minutes. Keep an eye on them, you want them to be finished cooking but not brown. Depending on your ovens heat power will depend if they need the full 15 minutes to bake.
Let cool completely before decorating with Royal Icing.