Using electric mixer, beat sugar and butter until well blended. Beat in flour, vanilla extract, and salt. Stir in oats. Cover and refrigerate cookie dough 1 hour.
Preheat oven to 350°F. Line 2 baking sheets with parchment paper.
Roll 1 tablespoon of dough into a ball. Place dough ball on baking sheet, spacing 3 1/2 inches apart (cookies will spread during baking). Repeat until sheet is full. Use the bottom of a drinking glass or a measuring cup to flatten cookies to 1 1/2-inch rounds.
Bake until cookies are golden brown, about 11 minutes. Let cookies remain on sheets 1 minute. Using spatula, transfer cookies to racks and cool completely. (Can be made 3 days ahead. Store cookies between sheets of waxed paper in airtight container at room temperature.)