Place the oven rack in the middle position and preheat the oven to 350 degrees F. Spray a 12 x 17.5 inch sheet pan (half sheet pan) with baking spray. Use your fingers or a pastry brush to spread it evenly over the pan. Place a sheet of parchment paper over the bottom of the pan and spray the top of the paper, spreading evenly with your fingers or the brush.

Whisk the cake flour, all-purpose flour, baking powder and salt together in a medium bowl. Set aside. Heat the milk and butter in a small saucepan over low heat until the butter is melted. Add the vanilla and set aside.

Separate 3 of the eggs and place the whites in the work bowl of a stand mixer. Using the whisk attachment, beat at medium-high speed until foamy. Gradually add 6 tablespoons of sugar and continue to beat until soft peaks form (do not over whip). Transfer whipped egg whites to a large bowl and set aside.

Add the yolks and remaining 2 large eggs to the empty mixer bowl along with the remaining 6 tablespoons of sugar. Using the whisk attachment, beat the egg and sugar mixture until thick and pale yellow in color, about 5 minutes. Transfer the yolk mixture to the bowl with egg whites.

Sift the dry ingredients over the top of the yolks and whites. Fold with a rubber spatula until the dry ingredients are just incorporated. Pour the milk mixture into the side of the bowl and fold until everything is incorporated.

Transfer the cake batter to the prepared pan and smooth out the top. Transfer to the oven and bake until the cake is set in the middle and beginning to turn golden brown, about 12 minutes.

Remove the pan from the oven. Immediately run a knife around the inside rim of the pan to loosen the cake. Flip the cake onto a tea towel that has been generously dusted with powdered sugar. Peel off the remaining layer of parchment and quickly roll the warm cake from one short side to the other inside the tea towel. Let the rolled cake rest and cool for 15 minutes.

Carefully unroll the cake and spread jam in a thin layer evenly over the top. Roll back up and transfer to a serving plate. Dust with powdered sugar, slice and serve.