Preheat your oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners or lightly grease them.
Brown the butter: Melt the butter in a small saucepan over medium heat until it turns golden brown and smells nutty. Remove from heat and let it cool a bit.
Mix wet ingredients: In a large bowl, whisk the brown sugar, eggs, vanilla, and milk. Stir in the slightly cooled brown butter.
Add dry ingredients: In another bowl, mix mochiko flour, baking powder, and salt. Slowly stir the dry ingredients into the wet mixture until smooth.
Fill muffin cups: Divide the batter evenly between the 12 muffin cups.
Bake for 45–50 minutes, or until the tops are puffed and golden, and the edges look crispy.
Cool and serve: Let the muffins cool in the pan for 10 minutes, then transfer to a rack to cool completely.