Beat the butter at medium speed with an electric mixer until creamy; gradually add the sugar, beating well. Add the eggs 1 at a time, beating until blended after each addition. Add the vanilla, beating until blended.
Combine the flour, baking soda and salt; gradually add the flour mixture to the butter mixture, beating at low speed until blended.
Divide the dough in half; wrap each portion in plastic wrap and refrigerate 1 hour.
Heat the oven to 350 degrees F. Line 2 baking sheets with parchment paper. Roll half of the dough out to 1/4-inch thickness on a floured surface. Cut out cookies with a 2 1/2-inch round cutter, and place 1 inch apart on the prepared baking sheets. Bake until the edges begin to brown, 10 to 12 minutes, rotating the sheets and switching their positions halfway through baking time. Let cookies stand on baking sheets 5 minutes. Transfer to wire racks to cool. Repeat procedure with remaining dough.