Cut parchment paper into 8 (12- x 6-inch) rectangles. Beat butter with a heavy-duty stand mixer on medium speed until creamy, about 2 minutes; add sugar, and beat until light and fluffy, about 3 minutes. Add egg and vanilla, and beat until combined.

Stir together flour, baking powder, and salt in a small bowl. Gradually add flour mixture to butter mixture, beating on low speed until blended.

Transfer dough to work surface, and divide into 4 equal portions. Place 1 dough portion between 2 parchment rectangles. Roll out dough to a 9- x 3 1/2-inch rectangle about 1/4 inch thick. Repeat with remaining dough portions and prepared parchment paper rectangles. Place dough rectangles (still between parchment paper) on a baking sheet, and freeze 30 minutes.

Pulse peach preserves in a food processor until large pieces are broken apart.

Remove dough from freezer. Remove top pieces of parchment from the dough rectangles. Spread about 2 1/2 tablespoons of the preserves over 1 dough rectangle. Sprinkle with about 8 teaspoons of the finely chopped pistachios. Top with 1 dough rectangle. Repeat layers with remaining preserves, pistachios, and dough rectangles, leaving the top rectangle uncoated. Trim dough stack to an 8 1/2- x 3 1/4-inch brick. Wrap dough in plastic wrap, and freeze 1 hour.

Preheat oven to 350°F. Remove dough from freezer, and slice into 1/4-inch-thick rectangles; place on parchment paper-lined baking sheets.

Bake in preheated oven until lightly browned around edges, 13 to 15 minutes. Cool on baking sheets 5 minutes. Transfer to wire racks, and cool completely, about 20 minutes.