Preheat oven to 325 degrees F. Butter the bottom and sides of a square 8-inch Pyrex baking dish.
Lightly whisk together the eggs, maple syrup, melted butter, and vanilla, just until incorporated. You don’t want too much air in the batter or it will end up cake-like and not dense and chewy as a brownie should be.
Using a fine-mesh strainer, gradually sift the cocoa powder into the mixture, stopping once in a while to whisk it in.
Add the coconut flour and the salt, whisking to combine.
Pour the batter into the prepared pan, using a rubber spatula to get it all out of the bowl. Bake until a toothpick inserted in the center comes out not wet but not completely dry, with a few moist crumbs, 40-45 minutes.
Cool the paleo brownies for 30 minutes, in the pan, before cutting into squares and serving.