Preheat the oven to 350°F (180°C).
Butter and flour three 8-by-2 inch (20-by-5 cm) round cake pans and line the bottoms with parchment paper. In a medium bowl or liquid measuring cup, whisk together the milk, egg whites, crème frâiche, and vanilla. In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, granulated sugar, baking powder, and salt. With the mixer running on low speed, add the butter one piece at a time, beating until the mixture resembles coarse sand, 3 to 4 minutes.
With the mixer still running on low speed, slowly add a little more than half of the wet ingredients. Increase the speed to medium and beat until the ingredients are incorporated, about 30 seconds. Return the mixer to low speed and 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.
Divvy the batter between the prepared pans and smooth the tops. Tap the pans gently on the counter, two times each, to help get rid of any bubbles.
Bake, rotating the pans halfway through, until the cakes are golden brown and a wooden skewer or toothpick inserted into the centers come out with a faint bit of crumbs, or a finger gently pressed into the top leaves a slight indentation, 28 to 30 minutes.
Transfer the pans to a wire rack and let cool for 30 minutes. Turn the cakes out onto the rack, remove the parchment paper, and let cool completely. Once cool, the cakes can be frosted or wrapped in plastic wrap and refrigerated overnight or for up to 1 week.