Boil water and pour some into a bowl to warm it. Set aside.
Pour ¼ cup more boiling water into a cup and let it cool slightly.
Sift the matcha through a fine-mesh sieve into a small bowl (optional).
Add 2 tablespoons of the hot water and stir, or whisk with a (bamboo) whisk until no lumps remain.
Stir in the remaining water and sweetener (optional) until dissolved. Pour it into the warmed bowl from step 1.
Heat the milk in a small saucepan over medium-low heat just until it begins to boil.
Froth the milk (optional).
Slowly pour the (foamy) milk into the bowl.
