In a small saucepan or microwave-safe bowl, warm 1 cup unsweetened plain plant milk with 3 tbsp vegan butter until the butter is melted. The liquid should be warm but not hotter than 110°F — aim for about 100–110°F. Use a kitchen thermometer to check.
If using active dry yeast: sprinkle 1 packet yeast over the warm milk mixture and let sit undisturbed for 10 minutes to bloom. If using instant yeast: stir the yeast directly into the warm milk mixture and proceed without the 10-minute bloom.
Transfer the milk/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. If the matcha is clumpy, sift it into the flour.
Add the flour mixture to the wet ingredients gradually, about 1/2 cup at a time, mixing after each addition. If using a stand mixer, use the dough hook on low speed as you add the flour. Continue until the dough comes together and is no longer easy to stir.
Turn the dough out onto a lightly floured surface and knead by hand until smooth and elastic, about 6–8 minutes. If using a stand mixer, continue kneading with the dough hook until the dough pulls away from the sides and forms a smooth ball.
Lightly grease a mixing bowl with a little melted vegan butter (you can use some of the 1/4 cup melted vegan butter listed for the filling). Place the dough ball in the greased bowl, cover with plastic wrap or a clean kitchen towel, and let rise in a warm place 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. Melt 1/4 cup vegan butter and set aside (reserve a little of the melted butter for greasing the pan if you like).
Preheat the oven to 350°F. Grease a 10x10-inch cast iron skillet or baking pan with some of the melted vegan butter.
When the dough has doubled, gently punch it down to release the air and transfer it to a lightly floured work surface. Roll the dough into a rectangle about 12 x 16 inches and roughly 1/6 inch thick.
Brush the rolled dough evenly with the melted vegan butter. Sprinkle the sugar-cocoa mixture evenly over the buttered surface, then scatter the chopped chocolate on top.
Tightly roll the dough lengthwise into a log. Using a serrated knife, cut the log into 1-inch slices (you should get about 12); cut thicker if you prefer 8 larger rolls.
Place the slices, cut side up, into the greased skillet or baking pan, leaving a little space between them for expansion. Cover loosely and let rest 10–15 minutes while the oven finishes preheating.
Bake the rolls at 350°F for about 30 minutes, until the tops are slightly golden.
While the rolls bake, make the matcha icing: in a bowl combine 1 cup powdered sugar and 2 tsp matcha powder. Whisk in plant milk 2 tbsp at a time (from the remaining plant milk) until the icing is smooth and reaches your desired consistency; use 2–4 tbsp total.
Remove the rolls from the oven and let them cool in the pan for 5–10 minutes. Drizzle or spread the matcha icing over the warm rolls and serve immediately.