Heat the oven to 350°F. Lightly coat a 9x13-inch baking pan with butter or baking spray.
Whisk together 4 large eggs, 2 cups sugar, milk, coconut milk, and 2 teaspoons vanilla extract in a large bowl until fluffy and pale golden in color.
In a sieve directly over the bowl with the liquid ingredients, sift together mochiko, 2 tablespoons matcha powder, 2 teaspoons baking powder, and ½ teaspoon sea salt directly into the liquid ingredients. Whisk until well incorporated. You do not need to worry about overmixing because mochiko does not have gluten.
Add the melted butter and mix until fully combined. The batter will be liquidy.
Pour into the Matcha Butter Mochi into the prepared baking pan. Lightly tap the pan against the countertop to level out the batter and pop any air bubbles. If you are using sesame seeds, evenly sprinkle sesame seeds in a thin layer over the top of the batter.
Bake the Matcha Butter Mochi for about 1 hour, or until the butter mochi no longer wiggles in the center when you shake the pan.
Place the pan on a wire rack and run a knife with a thin blade around the edge of the pan. Let the Matcha Butter Mochi cool in the pan completely, about 1 hour. Tip the Butter Mochi out on to a cutting board to slice and serve or wrap for storage.