Heat the oven to 350 degrees. Grease a half sheet tray and line with parchment or waxed paper. Sift all the dry ingredients 3 times to aerate. Cream the butter until smooth and fluffy in a mixer fitted with a whisk attachment; add 1 1/2 cups of the sugar and mix. Add the eggs, 1 at a time, mixing after each addition. Beat until fluffy and light.
With the mixer running at low speed, add 1/3 of the dry ingredients and mix. Add 1/2 of the buttermilk and mix. Repeat with remaining dry ingredients and buttermilk and mix until smooth. Pour the batter into the prepared pan and spread it out into the corners.
Bake until set in the center and springy, 20 to 25 minutes. Let cool in the pan on a wire rack. Lay a piece of parchment or waxed paper on a work surface. Turn the cake out onto the work surface and peel off the paper that the cake baked in. Flip the cake over again so the shiny top side is facing up.
Cut the cake in half crosswise so you have 2 rectangles of cake. Whip the cream with the remaining 2 tablespoons of sugar and vanilla until stiff. Spread an even layer over 1 piece of the cake, then flip the other piece over (so that the shiny side faces down, and will touch the cream) and place on top of the cream. Freeze for 1 hour.
Cut into bar-shaped sandwiches, (about 2 by 3 inches each). Wrap individually in parchment or waxed paper, as you would wrap a present, and serve frozen. The recipe can be made several weeks in advance and kept frozen, in a plastic container or individually wrapped, until ready to serve.