Preheat the oven to 350°F and place a rack in the center of the oven. Generously butter a 9X13 inch baking pan. Set aside.
In a large bowl, or in the bowl of a stand mixer, combine the mochiko, coconut milk, milk, sugar, melted butter, eggs, baking powder, salt and vanilla bean paste. Whisk together until smooth. Add the shredded coconut if you are using it and whisk it in. Let the batter sit for about 15 minutes at room temperature for the rice flour to fully absorb the liquid.
Pour into the prepared pan and bake for 1 hour 10 minutes to 1 hour 20 minutes, rotating the pan midway through the baking time, until the top of the cake is golden brown and the mochi feels firm when you press it in the middle. Remove from the oven and let cool in the pan on a wire rack.
Cut the mochi into small pieces about 2X2 inches square, and use a small spatula to remove them from the pan. Place each piece in a small muffin paper or on a piece of parchment or waxed paper.
Butter mochi can be stored in an airtight container at room temperature for up to 4 days. You can also freeze them; put them in a 325° F oven for about 20 minutes to revive and get a little chewy and caramelized on top.
