Arrange a shelf in the lower third of the oven, place a baking stone on it, and preheat the oven to 350 F/177°C.
Spray a 9 x 2-inch (23 x 5-cm) round cake pan with nonstick cooking spray and line with a parchment circle. (I very lightly spray the top of the parchment, too.).
In a mixer on medium speed, beat the butter to soften. Beat until it is light in color, about 3 minutes. Add sugar and continue to beat (cream) until very light, scraping down the sides and across the bottom of the bowl at least once. While creaming, feel the bowl; if it does not feel cool, place in the freezer for 5 minutes, then continue creaming.
Beat in the vanilla. On medium speed, blend in the oil.
On the lowest mixer speed, blend in the yolks, one at a time, mixing just to blend. Blend in the whole eggs, one at a time, mixing just to blend.
In a medium mixing bowl, beat flour, baking powder, and salt for 30 full seconds at medium speed with a hand mixer or with a fork or whisk by hand.
On the slowest mixer speed, blend over 1/2 of the flour mixture into the batter. Continue on slowest speed and blend in 1/2 of the buttermilk. Continue adding the remainder of the flour until all is incorporated. Blend in the remaining buttermilk.
In a cold bowl with cold beaters, whip cream until soft peaks form when the beater is lifted. Stir about 1/4 of the whipped cream into the batter to lighten. Then, fold the rest of the whipped cream into the batter.
Pour the batter into the prepared pan. Drop the pan onto the counter from a height of about 4 inches (10 cm) to knock out bubbles. Place the cake in the oven on the stone and bake until the center springs back when touched, or a toothpick inserted in the center comes out clean but moist, about 40 minutes. Ideally, the cake should not pull away from the sides until it has just come out of the oven. The center temperature should be about 209 F/98 C if you check by inserting an instant read thermometer.
Place in the pan on a rack to cool, about 10 minutes, then shake the pan to loosen the cake all around. Spray a cooling rack with nonstick cooking spray and invert the cake onto the rack to finish cooling. Peel off the parchment. Cool completely before slicing into layers, storing or icing.