Dissolve yeast and sugar in water and milk. Set aided for 10-15 minutes while yeast activates. Add in oil.
Combine flour, baking powder, baking soda, and salt. Stir in yeast mixture. Knead for 2-3 minutes and shape into a ball. Let dough rise for 2 hours in a warm place.
Divide dough into 14 - 16 equal size balls. Roll out each ball until a 1/4 inch think. Fold in half and place a small piece of wax paper in the middle. Let rest for another 30 minutes.
Fill wok or shallow pan with water, roughly 2 inches of water. Place buns in a lined bamboo steamer and add to the pan. Steam for 8-12 minutes.
For the Filling...
Start with a quick pickle on your carrots. Heat vinegar and water in a saucepan. Stir in sugar and salt until dissolved. Remove from heat and add carrots. Let sit for 30 minutes or store in a fridge for up to 3 days.
To prepare the mushrooms, heat oil in a frying pan and add mushrooms. Cook for 4-6 minutes until mushrooms have released their water and are nicely browned. Add garlic, soy, and mirin and cook for another 1-3 minutes. Toss with honey and butter and cook for another 1-2 minutes.
Assemble the buns by adding a small amount of cabbage and pickled carrot. Top with 2-3 slices of mushroom. Garnish with green onions and panko breadcrumbs.