Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking dish with butter or cooking spray.
In a large mixing bowl, whisk together the sweet rice flour, sugar, baking powder, and salt until well combined.
In another bowl, mix the coconut milk, whole milk, melted butter, ube puree, eggs, and vanilla extract until smooth.
Gradually pour the wet ingredients into the dry ingredients, stirring until just combined. Do not overmix; a few lumps are okay.
Pour the batter into the prepared baking dish and spread it evenly.
Bake in the preheated oven for 45-50 minutes, or until the top is golden brown and a toothpick inserted in the center comes out clean.
Allow the butter mochi to cool in the pan for about 10 minutes before slicing into squares. Serve warm or at room temperature.