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. Makes 3 dozen. Enjoy!