Put strawberries and sugar in a small saucepan. Heat over medium high heat, stirring occasionally, until strawberries have softened and a syrup has formed. Mash any remaining large strawberry chunks with a fork.
Meanwhile, heat water on stovetop until just about to boil. Strain matcha into a bowl and add water. Whisk or use a frother to completely suspend the matcha in water.
Divide the strawberry syrup between two glasses. Add ice cubes on top, filling almost to the rim.
Pour 1/2 cup milk into each glass.
Divide the dissolved matcha in two portions and pour over the milk in each glass. To create a layered effect, aim the matcha to hit the ice cubes rather than falling directly into the liquid.
Stir to mix before drinking.