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To make the starter: Weigh your flour for the starter; or measure it by gently spooning it into a cup, then sweeping off any excess. In a medium-large bowl or the bowl of a stand mixer, combine the flour, yeast, and water. Cover the bowl and let the starter rest for 1 hour, until it's bubbly.
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To make the dough: Weigh or measure the flour for the dough, then add it along with the rest of the dough ingredients to the bowl with the starter. Mix and knead the dough — using your hands, a stand mixer, or your bread machine set on the dough cycle — to form a sticky, elastic dough, 6 to 8 minutes. Resist the urge to add more flour; the dough is supposed to be soft and stick to the sides of the bowl.
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Cover the dough and let it rise for 1 to 1 1/2 hours, or until it's doubled in size; it may be a bit slow, so allow it plenty of time.
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To make the topping: In a medium bowl, combine the sugar, salt, vanilla, and butter until smooth. Stir in the flour in two additions, mixing to make a thick paste. (Note: You'll use the egg white later.) Cover the bowl and set the topping aside.
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Transfer the dough to a greased or floured work surface and divide it into 10 pieces (about 89g each). Roll each piece into a ball.
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Place the balls of dough onto a greased or parchment-lined baking sheet, leaving some space between them.
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To finish the chilindrinas: Divide the topping into 10 portions (about 33g each). Flatten each portion into a 3" to 4" disk by placing portions between two pieces of parchment or wax paper, and rolling with a rolling pin or flattening with your hands.
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Lightly beat the reserved egg white. Brush the top of each ball of dough with some of the beaten egg white, then add one of the disks of topping. Flatten the rolls slightly. (Reserve the beaten egg white.)
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Let the chilindrinas rise for 30 to 45 minutes, covered, until they're noticeably puffy. While the chilindrinas are rising, preheat the oven to 350°F with a rack in the center.
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Just before baking, brush the topping with beaten egg white and sprinkle sparkling sugar over each one.
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If you have a second baking sheet, nest the pan of chilindrinas into it; this extra layer of protection will help prevent the chilindrinas' bottoms from over-browning as they bake.
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Bake the chilindrinas for 18 to 24 minutes, until they're golden brown. Remove them from the oven and let them rest on the pan until they're cool enough to handle. Enjoy chilindrinas warm or at room temperature.
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Storage information: Store any leftover chilindrinas, well wrapped, for 3 days at room temperature; freeze for up to a month.