Preheat the oven to 375 degrees F. Butter an 8-inch-square baking pan and line with foil, leaving a 2-inch overhang on all sides; butter the foil.
Mix the applesauce, raisins, lemon zest and cinnamon in a bowl; set aside. Whisk the melted butter, milk and egg yolks in a shallow dish.
Dip half of the bread slices in the egg mixture and layer in the prepared pan, trimming as needed. Spread the applesauce mixture over the bread. Dip the remaining bread slices in the egg mixture and layer on top. Bake until the egg mixture is set and the bread is golden, about 35 minutes.
Meanwhile, beat the egg whites and confectioners' sugar in a bowl with a mixer on medium-high speed until stiff peaks form. Spread the apricot preserves on top of the cake. Spread the egg-white mixture on top of the preserves, forming peaks with the back of a spoon. Return to the oven until the meringue is golden, 5 to 7 minutes.
Let the cake cool in the pan, about 1 hour. Lift out of the pan using the foil and slice into squares.