Preheat oven to 350°F. In the bowl of a stand mixer, beat eggs until thick and bright yellow, then add the sugar and mix until combined.
In a separate bowl, sift together the flour, baking powder, and salt. Add the dry ingredients to the egg mixture alternately with the cream and vanilla, beating well.
Spread batter into two 8-inch round pans – it will seem thicker than usual cake batter, but that’s okay. Bake for 25 to 30 minutes, until a tester inserted in the center comes out clean.