Cream the butter and sugar with an electric mixer on high speed until fluffy and lighter in color. Stop mixer and add the egg, milk and vanilla. Mix at medium speed until combined well. (Don't be alarmed if it looks like it is curdling. It will recover when you incorporate the flour.)
Weigh or measure the flour and place in a medium bowl. Add the salt and combine with a whisk.
Gradually add the flour to the butter mixture in 2-3 batches on low speed and mix just until combined.
Divide dough into 3 portions. You can wrap each portion in plastic wrap, flattening it slightly and refrigerate until ready for use, at least one hour. (NOTE: If you want to make the cookies immediately, you can roll out the dough right away and place it on a cookie sheet in the refrigerator to chill. Preheat the oven while you are waiting. It will take about 15-30 minutes for it to be hard enough to use the cookie cutters.)
Preheat oven to 350°F (180°C). Remove one of dough disks from the refrigerator. Place on a lightly floured surface and use a rolling pin to roll to desired thickness. If the dough is very cold, you can let it warm up a tiny bit. (I do ⅛ inch but thicker is okay. The cookies will take longer the thicker they are.)
Use cookie cutters to cut out the dough shapes. Try to fit them like a puzzle to minimize dough wastage. (You can re-roll the dough, but those cookies will be a bit more tough.) Place cookies on a cookie sheet lined with parchment paper and add sprinkles if desired.
Bake for 7-10 minutes or until cookies are set and starting to brown a bit on the bottoms and edges. Cool for 5 minutes on the cookie sheet and then transfer to a wire rack to cool completely. (Plain cookies can then be iced.