Make the steamed buns: Combine active dry yeast, 1⁄3 cup warm water, and 1⁄4 tsp. sugar in a small mixing bowl and allow to bloom for 10 minutes. In a separate mixing bowl, combine dry milk powder, remaining sugar, vegetable oil, remaining water, and cake flour. Add bloomed yeast mixture and mix well. Cover with plastic wrap and allow to rise until doubled in size, about 45 minutes.
Add baking powder and salt and mix well. Allow dough to rest for 45 minutes.
Roll out dough to a thickness of 1⁄4". Brush surface lightly with vegetable oil and fold over once. Using a 2" ring mold, punch out buns. Do not reroll the dough. Allow the buns to rise until doubled in size. Steam until cooked through, about 4–7 minutes.
Make the duck laquer: Combine 1⁄2 cup water, glucose, 1⁄8 cup rice wine vinegar, black vinegar, star anise, red chile flakes, and ginger in a sauce pot and simmer for 20 minutes. Strain lacquer through a chinoise and chill. Reserve 1⁄8 cup for roasting the duck; cover and chill or freeze the remainder for another use.
Roast the duck: Preheat oven to 350˚. Season duck legs with salt and fresh ground pepper. Roast duck legs on a sheet tray lined with a rack, brushing with lacquer every 5 minutes, until cooked through and golden brown, about 20 minutes.
Make the pickled carrots: Combine 1⁄2 cup water, 3⁄4 cup rice wine vinegar, sugar, and 1 tsp. salt in a sauce pot and bring to a boil. Chill pickling liquid and pour over julienned carrots. Marinate carrots for at least 1 hour. Store chilled until ready to use.
To serve: Reheat steamed bao buns in stovetop steamer until just warmed through. Cut each bun in half and spread each side with Togarashi Mayonnaise. Top with 2–3 slices of roasted duck leg per bun, garnish with pickled carrots, and serve immediately.