Use higher fat, cultured butter for the cake topping, such as Plugra or Kerrygold, for richness. Based on a wonderful recipe by Maria del Mar Cuadra. Other stone fruits, such as plums or peaches, would be delicious here as well.
Adjust rack to the middle of the oven. Heat the oven to 350°F. Butter a 13x9x2 cake pan. Line the bottom of the pan with parchment, then butter the parchment. This will help ease the cake out of the pan.
To make the topping, in a stainless skillet or saucepan, melt 4 tablespoons of butter over medium heat. The butter will foam and then quickly the solids will brown. Swirl the pan as the butter cooks. When the solids are golden brown, about 5-7 minutes, remove the pan from the heat immediately. The butter can go from golden to burned quickly, so keep a close eye.
Stir the brown sugar, orange blossom water, vanilla, and salt into the butter. Sprinkle the wet, sandy mixture evenly over the parchment in the pan.
Halve the apricots, remove the pits, and cut each half into four wedges. Line the cake pan with the apricots in four long rows, nestling the apricots closely together.
To make the cake batter, sift the flour, baking powder and salt in a medium bowl. Set aside.
Using a stand mixer or hand mixer, in a large bowl beat the sugar and 8 tablespoons softened butter on medium high speed until light and fluffy, about 2 minutes. Add the egg yolks one at a time, beating well after each addition.
Alternate adding the flour mixture and the milk in three additions of each (six additions total). Add the vanilla and orange blossom water and mix until combined.
In another clean bowl with clean attachment, beat the egg whites until they hold stiff peaks, but are not dry. Take care not to over-beat the egg whites.
Using a rubber spatula, fold the egg whites into the batter until fully incorporated. Do not over-stir.
Scrape the batter evenly over the apricots in the cake pan and spread evenly.
Bake the cake until golden and the top bounces back when touched in the center, 45 to 50 minutes. Cool the cake for 15 minutes, then turn over onto a sheet pan or large serving platter. Remove the parchment if it came off onto the apricots. Serve immediately, or anytime for 3 days. Cover with plastic and keep at room temperature.