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To make the buns: In a medium bowl, whisk together the flour, sugar, yeast, and salt. In a small bowl or a liquid measuring cup, whisk together the milk, melted butter, and egg. Add the wet ingredients to the dry ingredients and mix to make a wet, sticky dough. Using a bowl scraper, transfer the dough to a lightly greased container, cover, and let rise at room temperature for 1 1/2 to 2 hours, until puffy and doubled in size, or refrigerate for up to 12 hours.
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Transfer the risen dough to a lightly floured work surface and knead gently until smooth, about 30 seconds. Divide the dough into 10 pieces (about 43g each). Shape each piece into a tight ball, sprinkling on more flour as necessary, and arrange, evenly spaced, on a parchment-lined baking sheet. As you shape, gently press down on each ball to flatten slightly.
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Cover the baking sheet and let the buns rise for 45 to 60 minutes, until puffy; they should spring back slowly when poked with a finger.
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Towards the end of the rise time, preheat the oven to 350°F with a rack in the center.
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Brush the risen buns with milk. Bake for 15 to 18 minutes, rotating halfway through, until the buns are evenly golden brown.
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Remove the buns from the oven and let cool completely on the baking sheet. (Buns can be made 1 day in advance; store airtight at room temperature.)
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To make the filling: In a small saucepan, whisk together the sugar, cornstarch, and salt. Using a rubber spatula, stir in the egg yolk until the mixture looks like coarse, wet sand. Pour in about half of the milk, whisking to break up any large clumps, then gradually stream in the remaining milk, whisking until the mixture is homogeneous. Set the saucepan over medium heat and cook, whisking frequently, until the pastry cream begins to thicken, about 4 to 6 minutes. Remove from the heat and whisk in the cold butter and vanilla until the mixture is silky and smooth.
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Transfer the pastry cream to a bowl or container, cover with plastic wrap or parchment directly touching the surface, and let cool to room temperature, 1 to 2 hours. (Pastry cream can be made up to 4 days in advance; store in an airtight container in the refrigerator. Bring to room temperature when ready to use.)
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To fill the buns: Insert a chopstick into the bottom of each bun, pushing it about halfway through. Wiggle it around assertively to make a 1/4"-diameter hole and form a cavity for the filling. Fit a pastry bag with a small round tip or snip off one corner from a resealable plastic bag to make a 1/4" opening.
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Whip the heavy cream to medium peaks. Scoop about one-third of the whipped cream into the pastry cream and whisk to combine. Gently fold in the remaining whipped cream until no streaks remain.
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Spoon the filling into the prepared pastry bag and pipe it into the buns, moving the tip around inside each bun as you work. Try to fill each bun with about 2 tablespoons (28g) of filling.
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To make the topping: Set a dry small saucepan over medium-low heat and sprinkle in about 1/4 cup (50g) of the sugar in an even layer. As the sugar begins to liquify, use a rubber spatula to encourage it to melt evenly by gently moving the liquid areas into the sugar crystals. When most of the sugar is melted, sprinkle more into the pan, repeating until all the sugar has been added.
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Continue to cook until the caramel is light amber, with no crystallized sugar remaining; when you tilt the pan, the caramel should move like syrup. (If the caramel is too hot, it will smoke and burn and move around the pan like water. If the caramel is too cold, it will stay thick and will be difficult to dip — adjust the heat accordingly.)
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To dip the buns: Take a bun in your dominant hand and hold it face down. With the heat under the caramel on the lowest setting, tilt the pan so the caramel pools to one side. Using tongs or your hands, quickly dip the top of the bun into the caramel. Lift it straight up, letting the excess drip back into the pan, then return it, caramel-side up, to the baking sheet to cool. Use caution when working with hot caramel; if it touches your skin, it’ll burn.
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After the caramel has cooled and set, smack the tops of the buns with the back of a serving spoon to shatter the caramel. Serve the buns immediately, when they’re at their best.
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Storage information: Filled buns can be refrigerated for several hours before they’re dipped in caramel. Dip in caramel right before serving — it will lose its crunchy texture as it sits.
