Preheat the oven to 350F with a rack in the middle. Line an 8x8 metal baking pan with foil, dull side up. Leave a couple inches overhang on two sides for easy removal. Lightly grease the foil.

Brown the butter: Place the cubed butter in a small, light-colored saucepan over medium-low heat. Once the butter has melted, turn the heat up to medium-high. Stir frequently with a heatproof spatula, scraping the sides and bottom of the pan as needed. The butter will crackle, foam, turn clear gold, then finally start browning. It’s done when the crackling subsides and you smell toasted nuts. This process takes about 10 minutes total, but the butter can go from browned to burnt in a flash—so keep an eye on it. Pour the butter and all the toasty bits into a glass measuring cup or medium bowl. Whisk in the coconut milk.

Mix the batter: In the bowl of a stand mixer fitted with the whisk attachment (you can also use a hand mixer or a whisk), combine the sugar, vanilla, salt, baking powder and eggs. Whisk on low to combine, then increase the speed to medium and beat until the mixture is thickened and pale, about 2-3 minutes (a little longer if by hand). Reduce the speed to medium-low and gradually stream in the butter-coconut milk mixture. Mix until smooth and combined, then scrape down the sides of the bowl. Sift in the mochiko and mix on low until the batter is smooth. Use a flexible spatula to fold from the bottom of the bowl a few times to make sure the batter is well-mixed and no pockets of flour remain. If there are any flour lumps, use the spatula to press them out.

Bake: Scrape the batter into the prepared pan (it will be on the thin side) and bake until the top is golden brown and feels dry and springy to the touch, about 55-65 minutes. Transfer to a wire rack and cool completely in the pan (preferably overnight). Use a sharp knife to cut into squares; wipe the blade clean with a warm towel between slices. Store in an airtight container at room temperature for up to three days.