**Make the Buns:**

1. – Mix the warm milk, sugar, yeast, flour, egg, salt, and ¼ cup softened butter in a large bowl until a shaggy dough forms.
2. – Turn the dough onto a clean surface and knead for 15 minutes. The dough will be sticky at first but will become smooth and stretchy. Do not add more flour.
3. – Alternatively, use a stand mixer with a hook attachment and knead for 10-12 minutes.
4. – Transfer the dough to an oiled bowl, cover with cling wrap, and let it rise in a warm spot until it doubles in size (about 90 minutes).

**Shape the Buns:**

1. – Punch down the risen dough and divide it into 8 equal pieces.
2. – Roll each piece into a tight, smooth ball.
3. – Flatten each ball into a disc (about 3.5 inches in diameter), place a cube of cold butter in the middle, and pinch the dough into a ball, ensuring the butter is well-sealed inside.
4. – Place the buns on a baking sheet lined with parchment paper, leaving at least 3 inches between each bun.
5. – Let the buns rise again for 30-45 minutes until puffed up.

**Prepare the Coffee Crust:**

1. – Whisk together the butter and sugar until light and fluffy. Add the egg and coffee mixture, and whisk again.
2. – Sift in the flour and cornstarch, then whisk until just combined.
3. – Transfer the mixture to a piping bag with a small opening.

**Bake the Buns:**

1. – Preheat the oven to 350°F during the last 20 minutes of the second rise.
2. – Pipe the coffee crust onto each bun, starting from the top and swirling outward.
3. – Bake for 18-20 minutes until the crust is light golden and crispy.
4. – Let the buns cool for 15-20 minutes before serving.