Combine butter and sugar in large bowl. Beat at medium speed, scraping bowl often, until creamy. Add 2 eggs and vanilla. Continue beating, scraping bowl often, until well mixed.
Reduce speed to low; add flour, 1 cup pistachios and salt. Beat, scraping bowl often, until well mixed.
Divide dough in half; wrap in plastic food wrap. Refrigerate until firm (at least 1 hour).
Heat oven to 350°F Roll out dough on lightly floured surface, one-half at a time (keeping remaining dough refrigerated), to 1/4-inch thickness. Cut with 2 1/2-inch round cookie cutter; cut each round in half. Place 1 inch apart onto greased baking sheets.
Beat together egg white and water in small bowl. Brush tops of cookies lightly with egg mixture; sprinkle with remaining chopped pistachios. Bake for 10 to 12 minutes or until edges are lightly browned.