Whisk 2 cups plus 2 tablespoons flour, coarse salt, and sugar in large bowl. Add butter. Using fingertips or back of fork, blend in butter until mixture resembles coarse meal. Add ice water by tablespoonfuls, tossing until moist clumps form. Gather dough into ball. Divide in half; shape each half into disk. Wrap in plastic. Chill at least 1 hour.
Line 2 large rimmed baking sheets with parchment. Working with 1 disk at a time, roll out dough on floured surface to about 13x11 inches. Trim to 12x10-inch rectangle, then cut into eight 5x3-inch rectangles.
Arrange 4 rectangles, spaced apart, on each sheet. Spoon 1 1/2 tablespoons preserves in row down center of each rectangle. Top preserves with second dough rectangle. Using fingertips, gently press all edges of each tart to seal; press all edges with tines of fork to double-seal. Using toothpick, poke a few holes in center of top dough rectangle. Cover; freeze tarts on sheets at least 2 hours and up to 1 week.
Position 1 rack in top third and 1 rack in bottom third of oven and preheat to 375°F Bake frozen tarts uncovered until golden, reversing sheets after 15 minutes, 25 to 30 minutes total (some preserves may leak out). Immediately transfer tarts to rack. Sift powdered sugar lightly over. Serve warm or at room temperature with fresh berries.