Preheat oven to 325°F (160°C). Grease an 8x8-inch baking dish, line the bottom with parchment paper, and grease the parchment as well.
Carefully separate the eggs into two bowls, ensuring no yolk gets into the whites. Let both sit at room temperature for 10 minutes.
Beat egg yolks, granulated sugar, and water on medium-high speed for about 3 minutes until pale, thick, and ribbon-like.
Add the cooled melted butter and vanilla extract, then beat on low speed for 30 seconds until just combined.
Add flour and a pinch of salt, mixing on low until just combined. Scrape down the sides of the bowl.
With the mixer on low, slowly pour in lukewarm milk and mix until smooth — the batter will be very thin, like crepe batter.
In a clean bowl with clean beaters, beat egg whites starting on low and increasing to high until stiff peaks form, about 3–4 minutes.
Gently fold egg whites into the batter in three additions using broad, sweeping motions — a few white streaks are okay, but fold until mostly incorporated.
Pour batter into the prepared dish and bake for 45–50 minutes, until the top is lightly golden and the center jiggles only slightly.
Cool completely in the pan for at least 2 hours (or refrigerate overnight), then dust generously with powdered sugar, slice into squares, and serve.