Preheat your oven to 350°F. Grease a 9x13-inch baking pan and line it with parchment paper, leaving an overhang on two sides for easy removal.
In a small saucepan, melt the butter over medium heat, swirling the pan occasionally. Continue cooking until the butter turns golden amber and smells nutty, about 5 minutes. Watch carefully to avoid burning. Remove from heat and let cool slightly.
In a large bowl, whisk together the mochiko sweet rice flour, granulated sugar, baking powder, and salt until evenly combined.
In a separate bowl, whisk together the coconut milk, eggs, browned butter, and vanilla extract until smooth. Pour the wet mixture into the dry ingredients and stir until a smooth, pourable batter forms with no lumps.
Fold in the sweetened shredded coconut until evenly distributed throughout the batter. Pour into the prepared pan and spread into an even layer.
Bake for 50 to 60 minutes, until the top is golden brown and the edges pull slightly away from the sides of the pan. A toothpick inserted into the center should come out clean.
Let the cake cool completely in the pan on a wire rack. Once cool, lift it out using the parchment overhang and slice into squares. The texture is best at room temperature, chewy and dense with crispy edges.