In bowl of a stand mixer, fitted with paddle attachment, beat butter until smooth, 30 seconds. Scrape down sides, add sugars and beat on medium-high speed until smooth and creamy, about 1 minute.
Add egg, vanilla and salt; beat to combine.
Scrape down sides of bowl, then add flour and mix on low until no pockets of dry flour remain.
Add desired mix-ins (*see notes) and mix just to combine.
Divide dough into two even halves, roll each into a 12-inch log on large pieces of plastic wrap. Wrap in plastic wrap and chill in refrigerator until firm, 2–4 hours.
Preheat oven to 350ºF (176ºC) with rack set in middle position.
Line two baking sheets with parchment paper.
Slice dough logs into ½-inch thick slices; arrange 12 slices on prepared baking sheets, about 1½ inches apart.
Bake cookies, one sheet at a time, until cookies just start to turn golden around the edges, 12-15 minutes. (Refrigerate the second baking sheet with cookie dough while the first is baking.)
Let cookies cool on sheet 5 minutes then transfer to a wire rack to cool completely.
Store cookies in airtight containers at room temperature for up to 1 week or store in the freezer for up to 1 month. Note: if you're making different flavors, store them separately from each other. If stored together, they will absorb flavors from one another.)