In the bowl of a stand mixer, add the flour, sugar, and yeast. Mix with a wooden spoon to combine.

Return the bowl to the stand mixer, attach the dough hook, and add the olive oil, egg, and water.

Mix on low speed to combine, about 1 minute. Add the salt, then knead on medium speed for 6–8 minutes, adding water 1 tablespoon at a time as needed, until the dough is smooth and slightly tacky.

Grease a large bowl with olive oil, add the dough, and cover with plastic wrap. Let it rise in a warm place until doubled in size, about 1–1½ hours.

While the dough is rising, heat 2 tablespoons of olive oil in a wide skillet over medium-low heat. Add the sliced onions and a pinch of salt.

Cook the onions slowly, stirring occasionally, for 25–35 minutes, until they are golden, soft, and sweet. Let cool completely before using.

Once the dough has doubled in size, punch it down and divide it into 10–12 equal pieces.

Flatten each piece into a circle about 3½–4 inches wide. Place 1 tablespoon of caramelized onion in the center of each circle.

Fold the edges of the dough up and over the onion filling, bringing them together in the center like you would close a dumpling. Pinch well to seal so no filling leaks out. Then flip the dough ball seam-side down, cup your hand gently over it, and roll it in small circles on the counter to form a smooth, tight, round bun.

Place the buns on a baking sheet lined with parchment paper, leaving 2 inches between each one. Cover with a kitchen towel and let rise again for 30–40 minutes, until puffy.

Preheat the oven to 380F and Brush each bun with the beaten egg.

Bake for 20–25 minutes, until deep golden brown. Let cool on the baking sheets slightly before serving.