Step 1: Make the Matcha Pastry Cream

1. Heat the Milk: In a medium saucepan, heat the milk over medium heat until steaming but not boiling.
2. Whisk the Eggs and Dry Ingredients: In a separate bowl, whisk together the egg yolks, sugar, cornstarch, and matcha powder until smooth and thick.
3. Temper the Eggs: Slowly pour half the hot milk into the egg mixture, whisking constantly to avoid curdling.
4. Cook the Cream: Return the tempered egg mixture to the saucepan with the remaining milk. Cook over medium heat, whisking constantly, until the mixture thickens and starts to bubble.
5. Add Butter and Vanilla: Remove from heat and stir in the butter and vanilla extract. Transfer to a bowl, cover with plastic wrap (pressed directly onto the surface), and refrigerate until completely cool.

Step 2: Prepare the Donut Dough

1. Activate the Yeast: In a small bowl, combine the warm milk and a pinch of sugar. Sprinkle the yeast on top and let sit for 5-10 minutes until frothy.
2. Mix the Dough: In a large bowl, combine the flour, sugar, and salt. Add the yeast mixture, melted butter, eggs, and vanilla extract. Mix until a shaggy dough forms.
3. Knead the Dough: Turn the dough onto a floured surface and knead for 8-10 minutes, or until smooth and elastic. Alternatively, use a stand mixer with a dough hook for 5-7 minutes.
4. Let It Rise: Place the dough in a greased bowl, cover with a damp towel or plastic wrap, and let it rise in a warm place for 1 hour, or until doubled in size.

Step 3: Shape and Fry the Donuts

1. Roll and Cut: Punch down the dough and roll it out to about 1/2-inch thickness. Use a 3-inch round cutter to cut out circles. Re-roll scraps as needed.
2. Second Rise: Place the dough circles on parchment-lined baking sheets, cover, and let rise for 30 minutes.
3. Heat the Oil: In a large pot or deep skillet, heat about 2 inches of oil to 350°F (175°C).
4. Fry the Donuts: Fry the donuts in batches, about 1-2 minutes per side, until golden brown. Use a slotted spoon to transfer them to a paper towel-lined plate to drain excess oil.

Step 4: Fill the Donuts

1. Prepare the Filling: Transfer the chilled matcha pastry cream to a piping bag fitted with a long nozzle or plain round tip.
2. Poke and Fill: Use a skewer or knife to poke a hole in the side of each donut. Insert the piping tip and fill each donut with the matcha cream until slightly puffed.

Step 5: Add Toppings

1. Powdered Sugar: Dust the filled donuts with powdered sugar for a classic look.
2. Matcha Glaze: If using glaze, whisk together the powdered sugar, matcha powder, and milk until smooth. Dip the tops of the donuts into the glaze and let set on a wire rack.