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Sift flour and salt together into a medium bowl. Beat butter, oil and sugar in a mixing bowl with an electric mixer on high speed until well blended, about 1 minute. With the mixer on low speed, beat in the flour mixture and vanilla until just combined.
Divide dough into 2 equal portions and roll each into a log about 6 inches long. Wrap in plastic wrap and refrigerate until firm, at least 1 hour and up to 1 day.
Preheat oven to 350°F. Line a large baking sheet with parchment paper or a nonstick baking mat.
Working with one log at a time, cut the cookies 1/4 inch thick. (To keep the cookies round, roll the log just slightly after each slice.) Place the cookies about 1 inch apart on the prepared baking sheet. Using a wooden skewer, make 4 holes in the center of each cookie so it resembles a button.
Bake until the cookies are pale golden brown, 9 to 11 minutes. Let cool slightly on the pan, then transfer to a wire rack to cool completely. Repeat with the remaining dough.