You will need fluted five-inch round tart pans for this recipe.
Make the pate brisee: Pulse flour, sugar, and salt in a food processor until combined. Add butter; process until largest pieces are about 1/2 inch. Add ice water; process until just incorporated (dough should not be wet or sticky), no more than 20 seconds. Squeeze a small amount of dough; it should just hold together. If not, continue to add water, 1/2 tablespoon at a time. Divide dough into thirds; shape into disks. Wrap each disk in plastic; refrigerate until firm, up to 2 days.
Turn out dough onto a lightly floured work surface. Roll out dough to 1/8 inch thick. Cut out three 7-inch rounds. Place dough in 3 fluted 5-inch tart pans with removable bottoms. Reroll remaining dough into an 8-by-12-inch rectangle; cut crosswise into scant 1/2-inch strips, and refrigerate strips until ready to use.
Preheat oven to 375 degrees. Make the filling: Stir together preserves and 1 tablespoon lemon juice. Spread 1/4 cup preserve mixture in bottom of each tart. Stir together raspberries, confectioners' sugar, cornstarch, lemon zest, and remaining 3 tablespoons lemon juice in a medium bowl. Spread raspberry mixture over preserves.
Arrange strips of dough in a lattice pattern over the top of each tart, trimming to fit as needed. Brush strips with egg. Place tarts on a rimmed baking sheet. Bake until filling is bubbling and crust is golden brown, 45 to 50 minutes.