Step 1

Pulse flour, granulated sugar, and 2 teaspoons salt in a food processor until combined. Add butter and shortening; pulse until mixture resembles coarse meal with a few pea-size pieces remaining. Add 3 tablespoons water and pulse. Add 3 tablespoons more and pulse. Next, work in longer pulses, stopping to scrape sides and bottom of bowl as needed, until dough holds together when pinched (if necessary, add remaining water, 1 teaspoon at a time). Shape into 2 rectangles.

Step 2

Roll out one rectangle between 2 sheets of lightly floured parchment to 12 by 10 inches, about 1/8 inch thick. Transfer on parchment to a baking sheet. Repeat with second rectangle; stack on top of first. Refrigerate both until firm, at least 30 minutes.

Step 3

Preheat oven to 375°F. Remove dough from refrigerator; set aside one sheet. Remove parchment from top of second sheet; spread jam evenly over dough, leaving a 3/4-inch border. Remove parchment from first piece of dough and place over filling. Fold edges in and press to seal with the tines of a fork. Poke top all over with tines of fork. Brush with beaten egg. Freeze 10 minutes or refrigerate 30 minutes. Bake until golden brown and bubbling in center, 40 to 45 minutes. Let cool on sheet on wire rack.

Step 4

Gradually whisk lemon juice into confectioners' sugar to create a glaze with the consistency of honey. Add remaining pinch of salt; whisk to combine. Spoon glaze over tart, spreading almost to edges; sprinkle with freeze-dried berries and let stand until set, at least 1 hour. Slice and serve, or store, lightly wrapped with foil, at room temperature up to 1 day.