In a large microwave-safe bowl, whisk together the mochiko, sifted cocoa powder, and sugar until no lumps remain.
Add the milk, melted butter, and vanilla extract to the dry ingredients. Whisk vigorously until you have a completely smooth batter with no streaks of flour.
Cover the bowl loosely with plastic wrap and microwave on high for 1 minute. Remove and stir thoroughly with a spatula.
Microwave for another 30 seconds, stir again, then repeat in 30-second intervals until the mixture transforms into a thick, glossy, stretchy dough (about 2 to 2.5 minutes total).
Dust a clean work surface generously with potato starch or cornstarch. Turn the hot mochi dough out onto the surface and let it cool for 10-15 minutes until just warm to the touch.
Coat your hands with starch, then divide the dough into 8-10 equal portions using a bench scraper or knife. Roll each piece gently between your palms to form smooth balls.
Roll each ball in additional starch to prevent sticking, then gently flatten into thick discs about 1/2 inch tall.
Let the mochi cool completely at room temperature for at least 20 minutes before serving to allow the texture to set properly.
