Mix wet ingredients: In a large mixing bowl, use an electric mixer to cream the butter and sugar until light and fluffy, about 2-3 minutes. Add the egg and vanilla extract and beat until combined. Pour in the sourdough starter discard and beat until the wet ingredients are incorporated.
Add dry ingredients: Add the flour and salt to the wet ingredients and use the electric mixer to incorporate them until just mixed. Use a spatula to scrape the sides of the bowl, making sure all of the ingredients have been combined. Divide the dough into 2 pieces.
Roll the dough: Shape the first piece of dough into a disc and place it between two pieces of waxed paper. Use a rolling pin to roll the dough between the two pieces of waxed paper until it is ¼" in thickness. (Repeat with the second piece.)
Chill the dough: Stack the sheets onto a baking sheet and place in the fridge for a minimum of 2 hours, or up to 48 hours.
Cut out cookie dough: Once you are ready to bake, preheat your oven to 350°F (176°C). Line a baking sheet with a silicone mat or parchment paper. Remove one sheet of dough from the fridge. Peel the bottom piece of waxed paper from the dough to release it. Place the sheet of dough back onto the waxed paper and remove the top piece of waxed paper. Cut out shapes using a cookie cutter and arrange them on the baking sheet. (Gather up scraps, roll between waxed paper and chill before cutting out.)
Bake: Bake the cookies for 11-12 minutes making sure not to over-bake. (The edges should be very light golden brown with hardly any color at all.)
Cool: Allow the cookies to cool on the baking sheet after you remove them from the oven for a minimum of 15 minutes. Transfer them to a cooling rack to cool completely before decorating.