Preheat the oven to 350°F. Grease two 9 x 13 inch aluminum pans with nonstick spray, line with parchment paper, then generously grease the parchment paper.
In a medium bowl, whisk together the flour, baking powder, and salt.
In the bowl of a stand mixer fitted with the whisk attachment, whip the eggs and brown sugar on low speed to combine. Turn speed up to high and continue to whip until the mixture is slightly more pale and very thick, 5 minutes. Scrape down the sides, then whip on low speed for 1 minute more.
With the mixer off, gently sprinkle a quarter of the flour mixture over the egg mixture, then mix on low speed just until incorporated. Try to avoid sprinkling the flour into the center of the whisk attachment space, where it will get "stuck" and not mixed in. Add the remaining flour in 3 discrete additions, turning off the mixer and scraping down the sides of the bowl each time.
With the mixture running on low, gradually drizzle the cream into the batter, adding the vanilla extract with the final splash of cream. Mix just until incorporated; do not overmix.
Pour the batter into the prepared pans, dividing it evenly (~1000 grams) in each, and use a spatula to smooth the tops into an even layer. Bake until the cake springs back lightly when touched in the very center, and a toothpick inserted into the center comes out clean, 30-40 minutes. Cool completely.