Preheat oven to 350°F. Line a 9x13-inch baking pan with parchment and lightly grease the paper and sides.
Separate the eggs while they are cold. Place the whites in a spotless bowl and the yolks in a larger mixing bowl; set both aside to come to room temperature for 15 minutes.
Sift the cake flour once and set aside.
Using an electric mixer, begin whipping the egg whites on medium speed. When they start to foam, add the cream of tartar and continue whipping. Gradually add 1/4 cup of the granulated sugar and beat until the whites hold firm but not dry peaks (stiff, glossy peaks). Transfer the meringue to a clean bowl.
To the egg yolks, add the remaining 3/4 cup sugar, the dark rum (or rum extract), and the kosher salt. Beat on high until the yolks become pale and thick and the mixture falls in a ribbon off the beaters, about 3–4 minutes.
Fold half of the sifted cake flour into the yolk mixture with a rubber spatula, then gently fold in half of the whipped whites. Repeat with the remaining flour and whites, folding carefully to keep the batter airy and light.
Quickly pour the batter into the prepared pan, smooth the top, and bake at 350°F for 20–25 minutes, or until the top is golden and a toothpick inserted near the center comes out clean. Avoid overbaking—this should be a soft sponge.
Remove the cake from the oven and let it cool in the pan on a wire rack until it is warm to the touch, about 30–40 minutes. Cooling in the pan keeps the cake from drying out.
While the cake cools, whisk together the coquito, rum extract, cinnamon extract, and freshly grated nutmeg in a pitcher until smooth. Chill the mixture if the coquito was not cold.
When the cake has cooled to warm/room temperature, use a skewer or fork to poke holes evenly across the surface. Pour about one-third of the coquito mixture over the cake, letting it settle and absorb for 10–15 minutes. Repeat with the remaining coquito in two more additions so the liquid soaks in gradually.
Cover the pan tightly with foil or plastic wrap and refrigerate for at least 8 hours or overnight so the tres leches soak fully into the sponge—overnight gives the best texture.
For the topping, chill a metal mixing bowl and beaters. Pour in the cold heavy cream, add 1/4 cup sugar, the rum extract and cinnamon extract, and whip on medium-high until soft peaks form, then beat to stiff peaks (about 2–3 more minutes). Do not overwhip.
Spread the whipped cream evenly over the chilled soaked cake at least 1–2 hours before serving. Dust lightly with ground cinnamon and scatter the toasted shredded coconut over the top. Return to the fridge until serving.
Slice into 10 pieces and serve chilled. Store leftovers covered in the refrigerator for up to 3 days.