Let the butter sit at room temperature until it's very soft. If your house is very cold, place it inside a zip bag, and carry it with you to warm up to room temperature, or place it near a window with direct sunlight. Do not melt it in the microwave. Let the egg sit out at room temperature.
The dough:
In a mixing bowl or the bowl of your stand mixer, add the softened butter and sugar. Cream the butter and sugar by mixing for 3 to 5 minutes until the butter has incorporated and the texture is airy and fluffy. I start on low speed and increase it as I go.
Add the egg and vanilla and mix until the egg has incorporated into the mixture and you have a smooth-looking mixture.
Make sure not to over-pack the flour when measuring. Add the flour and salt to the bowl, and mix just until the flour is mixed into the buttery mixture. Again, start on low speed and increase as you mix.
Cover the bowl with plastic wrap or remove it from the bowl onto a large piece of plastic wrap and wrap it tightly. Place it in the refrigerator for the dough to chill for 1 to 2 hours, up to overnight. You want the dough to be very firm.
The cookies:
Place the oven rack in the middle position if baking one cookie sheet at a time or position two racks as close to the middle. Preheat the oven to 350F. Line two baking sheets with parchment paper.
Remove the dough from the fridge. For easier rolling, divide the dough in half. I use a large knife and slice it in half.
Lightly flour a large board, mat, or counter surface and roll out the dough to ¼-inch to ½-inch (at most). Thicker cookies will take longer.
Cut the dough with cookie cutters. I start cutting from the ends in (or from one side to another) so it's easier to lift. Lift the cookies and place them on the prepared baking sheet. Gather the "trims" and roll them into a ball. Re-roll the dough and cut out the cookies as you go. See note below on sticky dough.
Baking:
One baking sheet at a time: bake the cookies for 10 to 12 minutes, remove them from the oven when the tops are golden. Two baking sheets at the same time: place one on the top of the two racks and the other directly below it. Set a timer for 7 minutes, then swap the position of the baking trays for the remaining 5 or so minutes.
Let the cookies on the baking tray sit on the counter for 10 minutes. Once they're cool to the touch, use a flat spatula to lift them from the baking sheet and place them on a cooling rack to cool fully before frosting. I often leave them on the counter overnight. I find they're easier to frost the next day when they're completely cool.
Notes
Yield: this recipe yields 24 large 2 ½-inch round cookies, more or less depending on the cut-outs you're using.
Room temperature ingredients and softened butter are essential to give the dough its texture. Cold eggs will yield a denser cookie.
Preheat the oven: don't insert the tray in the oven until it's fully preheated.
Sticky dough: If your dough feels sticky, sprinkle a little flour, form it into a ball, and re-roll. If it's still sticking, refrigerate the dough until firm. If you have kids "helping" make cookies, divide the dough into 4 or more pieces and keep them in the fridge, and grab them as you need them. When dough sits on the counter for too long, the butter softens, causing it to get sticky, making it hard to cut out and pick up.
Storage: cookies should be cooled completely and stored in an airtight container at room temperature for up to 7 days.
This dough freezes beautifully. I thaw it out in the fridge, usually overnight.