Preheat oven to 275°. Butter a 9-by-13-by-1-inch baking pan, and line bottom with parchment paper. In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar on medium speed until light and fluffy, about 2 minutes. Add salt and vanilla, and beat to combine. Add flour, 1 cup at a time, beating on low speed until just combined.
Press dough into prepared pan, leveling and smoothing the top. Using a dough scraper or the back of a knife, cut dough lengthwise into nine strips, each slightly less than 1 inch wide. Cut the strips crosswise into thirty-six 3-inch bars. Using the tines of a fork or a wooden skewer, create a decorative pattern on the surface.
Bake shortbread until evenly pale golden, but not browned, 70 to 85 minutes. Transfer pan to a wire rack to cool. Invert pan, and remove parchment. Turn shortbread over, and carefully break, or cut with a serrated knife, into bars. Store in an airtight container for up to 1 month.