Place the apricots, the sugar, and the apricot pits if desired in a medium-sized saucepan over medium heat. Stir to mix and cook the apricots until they are softened but still a pretty, vivid color, which will take 10 to 15 minutes. Stir them occasionally to make sure they don’t stick to the bottom of the pan. When they are cooked to your liking, remove from the heat, and taste for sweetness. Stir in additional sugar if desired. After about 15 minutes, remove the apricot pits.
Preheat the oven to 425 F (220 C). Line a baking sheet with parchment paper. Chill a large-size bowl and a whisk or whisk attachment or the whisk attachment for an electric mixer (for whipping the cream).
In a large bowl, or in a food processor, combine both flours with the salt, baking powder, and 4 tablespoons of the sugar. Stir well, or process once or twice, to mix. Add the butter and process, or cut it in with a pastry blender or two knives until it ranges from the size of coarse cornmeal to the size of a pea. Transfer the mixture to a large bowl, add 1 cup (250ml) of the cream all at once, and lightly mix until the dough gathers together but does not completely adhere. Be very careful not to overmix.
Turn the dough out onto a well-floured surface, dust it lightly with flour, and press it down with a rolling pin until it is somewhat rectangular in shape and about ½ inch (1.3 cm) thick. (You can also work on a piece of floured parchment paper). Fold it in thirds like a business letter, and turn it one quarter turn (90 degrees). Lightly flour it and gently press it out again until it is an even rectangle about ½ inch (1.3 cm) thick, fold it in thirds, and turn it another quarter turn. Lightly flour the dough again if necessary to keep it from sticking to the rolling pin, and roll it out to form a very even rectangle measuring 6 x 12 inches (15 x 30cm). Cut it in half, crosswise, then cut an X through each half, then cut a cross through each half so that you have sixteen triangular pieces. Sprinkle the pieces with the remaining 1 tablespoon of sugar. Transfer them to the prepared baking sheet, leaving about ¼ inch (1/2 cm) between them. Bake in the center of the oven until the shortcakes are golden and puffed, about 12 minutes. Transfer the shortcakes onto a wire rack to cool.
Whisk the crème fraîche in the chilled bowl until it holds a stiff peak. Whisk in the sugar and taste - add additional sugar if necessary.
When the shortcakes are cooled, arrange two pieces on the center of each of eight plates. Top with equal amounts of apricots, and a dollop of cream. Garnish with flower blossoms.