In a large mixing bowl, cream butter. Beat in 1/2 cup powdered sugar until fluffy. Add vanilla extract. In a medium bowl, combine flour, pudding mix and salt. Gradually mix flour mixture into wet ingredients until combined.
Divide dough in half and form into balls. Wrap in plastic wrap and refrigerate for at least 1 hour. Preheat oven to 350 degrees. Line baking sheets with parchment paper. Remove dough from refrigerator and make teaspoon size balls. Place on baking sheet and press down on balls slightly, spacing 1-inch apart.
Bake in preheated oven 9-10 minutes. Do not overbake. Remove from oven, and let them cool on baking sheet for 5 minutes. While cookies are still warm, toss in remaining powdered sugar. Set on wire rack until cooled completely. Makes about 3 dozen cookies.