For the matcha layer:
Combine sifted matcha and warm water in a small bowl and stir until it forms a smooth paste. You may need more warm water. Set aside.
Heat the milk in a small saucepan over low heat (do not boil). Whisk in the matcha paste (you don't have to add all the paste if you feel the matcha flavor is strong enough and the color is green enough) and the sugar. You'll probably notice little lumps of undissolved matcha - that's okay, mine didn't dissolve fully either. Remove from heat and let cool about 10 minutes. Stir in the heavy cream. Place in the refrigerator until completely cool, stirring every so often.
For the strawberry layer:
Puree strawberries and milk in a blender until completely smooth. Strain the mixture through a fine sieve to remove the strawberry seeds - you'll need to use a spoon to help push the mixture through the sieve. Whisk in the cream, sugar and vanilla until incorporated and the sugar has dissolved.
Freeze in alternate layers in your Zoku Quick Pop Maker (refer to freezing instructions that came with your Zoku if unsure), letting each layer freeze completely before adding the next layer. Or, fill molds with matcha latte layer, freeze for 30 seconds-2 minutes, depending on how thin or thick you want the outer layer to be. Stick a straw into the unfrozen liquid in the middle and suck it out. Fill the empty matcha shells with the strawberry layer and let freeze completely. Remove popsicles according to the Zoku instructions.