Preheat the oven to 220ºC/200ºC Fan/gas mark 7/425ºF.
Mix the flour, salt, baking powder and 3 tablespoons of the sugar in a bowl. Grate the butter into these dry ingredients and use your fingertips to finish crumbling the butter into the flour. Whisk the egg into the cream, and pour into the flour mixture a little at a time, using a fork to mix. You may not need all of the eggy cream to make the dough come together, so go cautiously.
Turn the dough out onto a lightly floured surface, and roll gently to a thickness of about 2cm / ¾ inch. Dip a 6½ cm / 2½ inch round cutter in flour and cut out as many rounds as you can. Work the scraps back into a dough, re-roll and finish cutting out you should get 8 in all.
Place the shortcakes about 2½ cm / 1 inch apart on a greased or lined baking sheet, brush the tops with the egg white, and sprinkle them with the remaining 2 tablespoons of caster sugar. If it helps with the rest of your cooking, or life in general, you can cover and refrigerate them now for up to 2 hours.
Bake for 10-15 minutes, until golden-brown, and let them cool for a short while on a wire rack. Meanwhile, crush half the strawberries with the spoonful of sugar and the few drops of balsamic vinegar if using, and halve or quarter the remaining strawberries, depending on their size. Whip the double cream, if you're using
The shortcakes should be eaten while still warm, so split each one across the middle and cover with a spoonful of the crushed strawberry mixture, a few halved or quartered strawberries, then dollop some whipped cream or creme fraiche on top, and set the top back on.