Preheat the oven to 350°F. Line an 8 by 8-inch pan with parchment paper with excess on all sides to create 'wings' that can be used to pull the brownies out later.

To make the brownie layer: In a saucepan, combine the butter and brown sugar and cook over medium heat, stirring a few times, until the butter starts to simmer. Remove from the heat and add the chocolate, stirring until the chocolate has melted. Pour into a bowl and set aside until cool enough to touch, 5 to 10 minutes.

To the bowl, add the eggs and vanilla and whisk together until the mixture is shiny and smooth, 2 to 3 minutes. Add the cocoa powder, flour, espresso powder, salt, and baking soda and mix with a spatula until all the flour has been incorporated. Pour the batter into the lined pan and use the spatula to smooth the surface. Rap the pan on a kitchen towel set on the counter a few times to get it evenly distributed.

Bake until the edges and top are set, 22 to 25 minutes. Set on a rack to cool.

To make the haupia layer: In a saucepan, whisk together the granulated sugar, cornstarch, and salt. Whisk in the coconut milk and whole milk and set over medium heat. Whisking constantly, cook until thickened, 6 to 8 minutes.

Immediately pour the haupia mixture over the brownies in an even layer, quickly smoothing the top with an offset spatula. Rap the pan on a kitchen towel set on the counter a few times to get it evenly distributed.

Refrigerate until the haupia is completely set, at least 2 hours. Cut into 16 squares and serve chilled.