Have a small stainless steel bowl near the stove. Melt butter in a medium saucepan until it bubbles. Continue to cook, whisking gently, until the butter is golden brown and the milk particles suspended in it are reddish brown. Immediately pour the brown butter into a small pot or microwavable bowl to stop the browning.

Position a rack in the lower third of the oven and preheat the oven to 350°F. Line the bottom of 2 or 3 9-inch or 8-inch respectively pan with parchment paper, but do not grease the sides of the pan.

Have a 4- to 5-cup bowl ready to pour it into as well— the bowl must be big enough to allow you to fold some batter into the butter later.

Press the cocoa through a fine strainer into a medium bowl. Add the flour and whisk to blend.

Combine the eggs, sugar, and salt in the bowl of the stand mixer and beat with the whisk attachment on high speed for 6 to 7 minutes. The mixture should be light colored and tripled in volume, and you should see well-defined tracks as the whisk spins; when the whisk is lifted, the mixture should fall in a thick, fluffy rope that dissolves slowly on the surface of the batter.

Just before the eggs are ready, heat the brown butter until very hot and pour it into the reserved bowl.

Remove the bowl from the mixer. Sift one-third of the flour over the eggs. Fold with a large rubber spatula until the flour is almost blended into the batter. Repeat with half of the remaining flour. Repeat with the rest of the flour. Scrape about one-quarter of the batter into the hot butter. Fold until the butter is completely blended into the batter. Scrape the buttery batter over the remaining batter and fold just until blended. Scrape the batter into the pan.

Bake for 15 to 20 minutes, rotating the pans from back to front and upper to lower about halfway through the baking time, until the cakes have puffed up and then settled level but haven’t pulled away from the sides of the pans and a toothpick inserted in the center comes out clean. Set the pans on a rack. While the cake is still hot, run a small spatula around the inside of the pans, pressing against the sides of the pan to avoid tearing the cake.

Invert the pan onto the rack to remove the cake and peel off the parchment liner. The cake should be completely cool before filling. The cake may be wrapped airtight and stored at room temperature for 2 days, or frozen for up to 3 months.

To make the filling, in a heavy nonreactive saucepan, whisk the egg yolks with the milk, sugar, salt, and vanilla. Add the butter. Cook over medium heat, stirring constantly with a silicone spatula, scraping the bottom, sides, and corners of the pot.

When the mixture starts to boil, adjust the heat so that it boils actively but not furiously. Cook until the mixture is golden brown and thickened, about 12 minutes total cooking time. Take the pan off the heat and stir in the coconut and pecans. Cool completely before using.

To assemble the cake, if using 3 cakes, set one cake layer on a serving platter. Spread with one-third of the filling. Top with a second cake layer. Spread with half of the remaining filling. Put the third cake layer on top and cover it with the remaining filling. If using two cakes, use half the filling in between and half on top. Leave the sides exposed. Serve at room temperature.