Warm 1 cup unsweetened plain plant milk with 3 tbsp vegan butter in a small saucepan over low heat or in the microwave until the butter is melted and the liquid reaches 100–110°F (use a kitchen thermometer). Do not exceed 110°F.
If using active dry yeast: sprinkle 1 packet yeast over the warm liquid, stir gently, and let sit 10 minutes until foamy. If using instant yeast, you may skip this proofing step and add the yeast directly to the dry ingredients in step 5.
Transfer the foamy yeast mixture to a large mixing bowl or the bowl of a stand mixer. Stir in 1 tbsp cane sugar and 1/4 tsp salt.
In a separate bowl, whisk together 3 cups unbleached all-purpose flour and 1 tbsp matcha powder; sift the matcha into the flour if it is clumpy.
Add the flour mixture to the wet ingredients in 1/2-cup increments, stirring with a wooden spoon or mixing on low with the stand mixer (use the dough hook). Continue adding and mixing until the dough comes together and pulls away from the sides of the bowl.
Turn the dough out onto a lightly floured surface and knead by hand until smooth and elastic, about 5–8 minutes. If using a stand mixer, continue kneading with the dough hook until the dough is smooth and forms a ball (no additional hand kneading needed).
Place the dough ball in a lightly greased bowl, cover with plastic wrap or a clean kitchen towel, and let rise in a warm, draft-free spot until doubled in size, about 1 hour.
While the dough rises, mix the filling: combine 1/4 cup cane sugar and 1 tbsp cocoa powder in a small bowl. Finely chop 2 ounces vegan baking chocolate and set aside. Also melt 1/4 cup vegan butter and keep it ready for spreading.
Preheat the oven to 350°F. Grease a 10x10-inch cast iron skillet or baking pan.
When the dough has doubled, punch it down to release the air and turn it out onto a floured work surface. Roll the dough into a rectangle about 12 x 16 inches and approximately 1/6 inch thick.
Brush the rolled dough evenly with the 1/4 cup melted vegan butter. Sprinkle the sugar–cocoa mixture evenly over the buttered surface, then scatter the chopped vegan chocolate on top.
Starting at one long edge, roll the dough tightly into a log. Pinch the seam to seal. Using a serrated knife, cut the log into 1-inch slices (you should get about 12 slices; cut thicker if you prefer 8 larger rolls).
Arrange the slices cut-side up in the prepared 10x10-inch skillet or baking pan, leaving a little space between them for expansion.
Bake for 30 minutes, until the buns are puffed and slightly golden on top.
Meanwhile, make the matcha icing: in a bowl, whisk together 1 cup powdered sugar and 2 tsp matcha powder. Add 2 tbsp plain unsweetened plant milk and whisk until smooth; add more plant milk, 1 tbsp at a time, up to 4 tbsp total to reach your desired consistency.
Remove the buns from the oven and let them cool for 5–10 minutes. Drizzle or spread the matcha icing over the warm buns and serve immediately.