Position the rack in the center of the oven and preheat to 400 degrees F. Spray a baking sheet with vegetable-oil cooking spray.

To make the shortcakes:
Sift together the flour, the 1/4 cup plus 1 tablespoon granulated sugar, the baking powder, and kosher salt in a medium bowl. Transfer to the bowl of a food processor fitted with the steel blade. Add the butter and pulse 3 or 4 times at 2-second intervals until the dough resembles coarse meal. Add the cream and process until the dough begins to come together, a few seconds more. It will still be crumbly at this point. Do not overwork or the shortcakes will be tough.

Transfer the dough to a well-floured work surface. If the dough is sticky, sprinkle it with more flour. Using a lightly floured rolling pin, roll out the dough into a rectangle about 12 inches long. Fold the left end to the middle and the right end up over the left end to form 3 layers. Roll out the dough to 3/4-inch thickness. Using a 3-inch-wide heart-shaped cutter, cut into 3 individual cakes. Gather the scraps, roll out the remaining dough, and cut out 3 more hearts.

Place the shortcakes at least 2 inches apart on the prepared baking sheet. Brush the tops lightly with the beaten egg mixture and sprinkle with the 2 teaspoons granulated sugar. Bake until the shortcakes are light golden brown, 15 to 20 minutes.

Meanwhile, prepare the strawberries:
Stir together the strawberries, raspberry preserves, granulated sugar, and lemon juice in a medium bowl. Put 1 cup of the mixture in a small bowl and mash with a fork or potato masher. Add to the strawberry mixture and set aside.

To make the whipped cream:
Combine the heavy cream, sour cream, granulated sugar, and vanilla in the bowl of a stand mixer fitted with the whisk attachment. Beat on high speed until soft peaks form, about 1 minute.The cream should slowly fall over forward when lifted with a whisk. Alternatively, use an electric hand mixer.

To serve, using a serrated knife, cut 2 shortcakes in half horizontally and place the bottom halves on each of 2 plates.Top with the strawberries, dividing evenly. Add a dollop of whipped cream and top with the remaining shortcake halves. Dust with powdered sugar and garnish with the mint sprigs.

Do-Ahead:
These shortcakes freeze beautifully. Freeze the uncooked shortcakes on a baking sheet as directed but do not bake. Instead, place the baking sheet in the freezer, then transfer the frozen cakes to a sealable plastic bag. When ready to bake, preheat the oven to 400 degrees F and bake for 25 to 30 minutes. To prevent the bottom of the cakes from browning, put the frozen shortcakes on a baking sheet stacked atop another sheet, place in the top third of the oven, and bake for 15 to 20 minutes.

When we entertain, we whip the cream before our guests arrive, refrigerate it, and whisk to the desired consistency just before serving. We also like to warm the shortcakes just before serving.