Adjust an oven rack to the middle position. Preheat the oven to 350F (180C). Butter and flour a half sheet pan (12 x 16 inches, with a 1-inch rim) and line the bottom with a piece of parchment paper.
In a medium bowl or liquid measuring cup, whisk the eggs, egg yolks, vanilla, creme fraiche, and milk.
In the bowl of a stand mixer fitted with a paddle, mix the flour, sugar, baking powder, baking soda, and salt on low until combined. With the mixer running on low, add the butter one piece at a time, beating until the mixture resembles coarse sand. With the mixer still running on low, slowly add half the wet ingredients. Increase the speed to medium and beat until incorporated, about 30 seconds. With the mixer running on low, add the rest of the wet ingredients, mixing until just combined. Increase the speed to medium and beat for 20 seconds (the batter may still look a little bumpy). Scrape down the sides and bottom of the bowl, and use a spatula to mix the batter a few more times.
Pour the batter into the prepared pan and smooth the top. Tap the pan gently on the counter 2 or 3 times to help get rid of any bubbles. Bake 15-18 minutes, rotating the pan halfway through, until the cake is golden brown and pulls slightly away from the sides and a wooden skewer or toothpick inserted in the centers comes out clean.
Transfer the pan to a wire rack and let cool for 30 minutes. Carefully turn the cake out onto a large piece of parchment paper, remove the other parchment paper from the bottom of the pan, and let cool completely. Cake can be wrapped in plastic and refrigerated overnight, or frozen for up to 1 week.