In a food processor, pulse together the flour sugar and salt to blend. Add in the butter slices and process into a coarse meal, as you would for a pie crust or shortbread. If you do not have a food processor you can do this by hand using a pastry blender.
In a separate bowl, whisk the egg, vanilla, lemon peel, and lemon juice to blend. Add mixture to food processor or bowl with flour-butter mixture; whirl or stir until the dough forms a ball.
Divide dough in half. Stir fruit and nuts into one half of the dough. Roll each half between sheets of cooking parchment into an even 8 by 16 inch rectangle, then stack the rectangles and chill until firm, about 2 hours. If this is too large to place in the fridge, roll out two 8 inch squares of each dough for a total of 4 sheets.
Peel off the top sheet of parchment from each dough rectangle. Brush the plain dough very lightly with water. Place the sheet of fruit dough over the plain dough. Lightly press together to seal. Remove the top sheet of parchment. Trim the edges even. If necessary, let the dough warm just until pliable. Beginning with a long side, roll into a cylinder, pulling paper from dough as you roll. Discard parchment. If you rolled a full sized sheet, cut cylinder in half crosswise. Wrap each half in plastic wrap and freeze until firm, about 30 minutes, or up to 2 weeks.
Pre-heat oven to 350 degrees.
Unwrap half the dough at a time. Using a thin, sharp knife, cut into 1/8-inch-thick rounds. Place 1/2 inch apart on cooking parchment baking sheets.
Bake cookies until pale golden, 12 to 14 minutes. Transfer cookies to racks to cool completely.