Perfect Berry Shortcakes

Adjust oven rack to lower-middle position and preheat to 425 degrees. Mix flour, salt, baking powder and 3 Tbs. sugar in a medium bowl. Grate 2 Tbs. of the butter on the coarse holes of a box grater into dry ingredients; toss to coat. Repeat grating and tossing with remaining butter. Combine egg and half-and-half; pour into flour mixture. Toss with a fork to form large clumps. Lightly press clumps into a ball; add a teaspoon more half-and-half to the bowl if dough won't come together.
Turn dough onto work surface; press into an 8-by-4- to 5-inch rectangle. Cut into 6 squares, placing them 1 inch apart on a small baking sheet. (Can be refrigerated up to 2 hours before baking.) Before baking, brush tops with optional egg white for a particularly attractive sheen. Sprinkle with remaining 2 Tbs. sugar. Bake until golden brown, about 12 to 14 minutes. Let cool until warm, 5 to 10 minutes.
Mix thawed and fresh berries with sugar in a bowl; let stand until sugar dissolves. With a hand mixer, beat cream to soft peaks, gradually adding 1 Tb. sugar, then vanilla.
Split each cake crosswise; spoon a portion of berries over each cake bottom, then a dollop of whipped cream over berries. Cap with cake top and serve immediately.