Beat the softened butter with the sugar for 20 min. Add the yolks 1 by 1 and beat the mixture for another 10 min.
Beat the egg whites to a thick snow. Pour half of the flour at once into the mixture and stir. Add the other half spoon by spoon, while alternating with the beaten whites, be sure to stir very carefully.
Pour the dough into a buttered and floured rectangular oven dish in a layer about 3/4″ (2 cm) thick. Wash the apricots, dry them, cut them into halves, remove the pits and arrange on the dough with the cut side facing up, about 1 1/2″ (4 cm) from one another.
Bake the cake in a moderate oven to a pale pink. Once it cools, cut it into square pieces in such a way so that there's an apricot in the middle of each. Sprinkle with powdered sugar.