Preheat oven to 350F.
Butter 3 heavy large baking sheets. Using electric mixer, beat 2 cups butter and 1 cup powdered sugar in large bowl until light and fluffy. Beat in vanilla and salt, then pistachios and cherries. Using spatula, stir in all flour (do not overmix dough).
Shape dough by generous tablespoonfuls into footballshaped ovals. Place on prepared sheets, spacing 1 inch apart. Bake cookies, 1 sheet at a time, until bottoms just begin to color, about 16 minutes. Cool cookies on sheets 10 minutes before coating.
Pour generous amount of powdered sugar into medium bowl. Working with 5 or 6 warm cookies at a time, add cookies to bowl of sugar; gently turn to coat thickly. Transfer cookies to sheet of waxed paper. Repeat to coat cookies with sugar again; cool completely. do ahead Can be made 4 days ahead. Store airtight at room temperature.
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