Preheat oven to 350 degrees, with racks in upper and lower thirds. In a large bowl, using an electric mixer, beat butter and sugar until light and fluffy; beat in vanilla and salt. With mixer on low, gradually add flour, beating just until combined. Fold in pecans.
Roll dough into 1 1/2-inch balls, and place on two baking sheets, 2 inches apart. With the dampened bottom of a glass, lightly flatten each ball.
Bake until cookies are golden brown, 15 to 17 minutes, rotating sheets halfway through. Transfer to wire racks, and let cool.
For best results, line cookie sheets with parchment prior to baking.