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Step 1
Add the sugar and yeast to the warm water in a large liquid measuring cup or mixing bowl. Stir and let sit for a few minutes to get foamy. Add the oil.
Step 2
In a large mixing bowl whisk together the flour, baking powder, and salt. Add the yeast mixture and mix together with a wooden spoon or by hand (wetting your hand to prevent sticking), until you have a coarse dough. Turn the dough onto a lightly floured counter and knead by hand around 5 minutes until the dough is smooth and elastic. Add flour if necessary, though give it a minute before adding as it takes time for the flour to absorb the moisture. You should have a soft and lightly tacky dough (not sticky). Alternatively you can knead in a stand mixer on low speed for 4-5 minutes.
Step 3
Transfer the dough ball to a lightly oiled mixing bowl and cover with plastic wrap. Let rise for 45 minutes - 1 hour until doubled in size.
Step 4
Trim the ends of the eggplant and cut them in half. Brush with oil, then grill over medium-high heat (using grill pan or outdoor grill) until the eggplant is tender when pierced with a fork, about 4-5 minutes each side, but this depends on size of eggplant. When cool enough to handle, cut grilled eggplant into ~3/4" cubes
Step 5
Heat the oil and garlic in a small pot over low heat. When the garlic turns a pale golden--not too dark or it will taste burnt--carefully add the liquids and whisk to combine. Bring to a simmer and let simmer for about 5 minutes, stirring occasionally until the sauce is thickened.
Step 6
Toss the eggplant in the sauce.
Step 7
Punch down the dough and divide into 12 (~54 g) balls of dough. Let rest a few minutes before shaping so the gluten has a chance to relax.
Step 8
Prepare your steaming apparatus and bring water to a boil. Line the steamer with parchment to prevent sticking and/or cut 12 ~2.5" squares of parchment paper to use with each bun.
Step 9
Pat or roll out each piece of dough into a ~5" round. Place in your non-dominant hand, and a heaping tablespoon of the eggplant to the center. Take opposing sides of the dough and pinch together over the filling with your thumb and forefinger. Repeat pinching the dough together as you go around the bun until all of the dough meets in the center. Pinch and twist the top to seal it off. Place the bun on a piece of parchment paper and repeat with the remaining dough.
Step 10
When the water is boiling place the buns in batches in the steamer basket. Be careful not to overcrowd as they double in size while steaming. In my steamer I can make 4 at a time. Keep the remaining shaped buns in the refrigerator while you steam so that they do not get over proofed.
Step 11
Steam each batch 12-15 minutes. Best enjoyed warm or on the same day. Store leftovers in a tightly sealed container in the fridge and reheat in the microwave until warm. Cover with a damp paper towel while microwaving if they feel dry.