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To make the dough: Place the cardamom pods in a mortar and use the pestle to crush them slightly; discard the green shells. Use the pestle to coarsely grind the seeds, like cracked black pepper. Measure 1 teaspoon of the coarsely ground seeds and add to the bowl of a stand mixer. (See “tips” below for information on using pre-ground cardamom.)
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Add the milk, sugar, yeast, and salt to the mixer bowl and whisk until well-combined. Measure 1 1/2 tablespoons (25g) of the beaten egg and add to the mixer. (Reserve the remaining egg for the egg wash.)
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Fit the mixer with the dough hook attachment. Add the flour, then the butter, and knead on low speed until the dough comes together, about 2 minutes. Scrape the bowl, then increase the speed to medium-high and knead for 8 to 10 minutes, until the dough is smooth, cleans the sides of the bowl, and passes the windowpane test.
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Use a bowl scraper to gather the dough into a ball, then cover the bowl, and let rise until puffy and doubled in size, 1 1/2 to 2 hours. After the dough has risen, you can cover the bowl tightly and store the dough in the refrigerator for up to 24 hours or proceed with the next step.
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To make the filling: In a small bowl, mix all of the ingredients together until smooth. Cover and set aside at room temperature.
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To make the custard: In a small pot, bring the milk and the seeds of half a vanilla bean to a bare simmer. (If using vanilla bean paste, you’ll add it in the next step.)
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In a small bowl, whisk the sugar and flour until combined. Add the yolks and vanilla bean paste, if using, and whisk vigorously until lightened in color.
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While whisking, gradually pour the hot milk into the egg mixture. Once you’ve added about half of the milk, return the egg mixture to the pot, still whisking as you pour. Return the pot to the stove over medium heat and cook, whisking constantly and getting into the corners of the pan, until the custard thickens to a loose pudding consistency, about 2 minutes. Transfer to a bowl and let cool, stirring occasionally. (Custard can be made 3 days in advance. If making ahead, place a piece of plastic wrap or parchment on top of the pastry cream so that it touches the surface, wrap the bowl well, and refrigerate until ready to use.)
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To shape and bake the sun buns: Line a baking sheet with parchment.
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Transfer the risen dough to a lightly floured surface. Roll into a 12" square about 1/4" thick. Spread the filling over about half of the square (no need to leave a border), then fold the bare side over the top to make a rectangle about 12" x 5 1/2". To make shaping easier, slide the dough onto a sheet of parchment and refrigerate for 20 to 30 minutes, until the butter is slightly firm. (If you refrigerated the dough overnight, you can skip this step if the dough still feels cool.)
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Gently roll out the dough to extend the width to slightly over 6", then trim the dough along the folded edge to expose the filling. Reserve the trimming.
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Using a sharp knife or a pizza or pastry wheel, cut the dough into eight pieces, lengthwise, to make long 3/4"-thick strips.
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Working with one piece at a time, twist the ends of each dough strip in opposite directions.
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Gently coil the twisted rope into a single-layered spiral. Carefully transfer to a baking sheet, tucking the tips of the spiral underneath itself. As you shape, incorporate the thin trimming from step 11 into whichever dough strip is smallest.
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Once all the sun buns are shaped, cover the baking sheet and let them rise at room temperature for 1 to 2 hours, until they spring back slowly when gently poked. Toward the end of the rise, preheat the oven to 400°F with a rack in the center.
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Using floured fingers or a lightly greased spoon, make an indentation in the center of each bun about 3/4" to 1" wide.
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Spoon a heaping tablespoon (about 20g) of custard into the indentation in each bun (a Tablespoon Scoop works well here). Brush the exposed dough with the reserved beaten egg.
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Bake for 18 to 20 minutes, until the custard is set and the buns are deep golden brown. (A little bit of the filling may leak onto the parchment during baking; that’s OK).
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Remove the sun buns from the oven and let them cool slightly on the baking sheet before transferring to a wire rack. Serve the sun buns warm or at room temperature.
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Storage information: Store leftovers in the refrigerator, well-wrapped, for several days. Reheat at 400°F for 10 minutes before serving.
