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To make the dough: In the bowl of a stand mixer fitted with the flat beater attachment, combine the flour, sugar, yeast, and salt. Add the butter and mix on low speed until the butter is about the size of peas and the mixture resembles coarse meal, 3 to 4 minutes.
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In a small bowl, whisk together the egg, vanilla, and milk.
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Add the egg mixture to the bowl of the stand mixer. Mix on low until the liquid is nearly all incorporated, about 30 seconds. Transfer the dough to a lightly floured surface and finish working the dough by hand, sprinkling more flour as necessary, until you have a homogenous dough that's soft and slightly sticky but still manageable; you might still see pieces of butter, which is OK. Form the dough into a 1"-thick rectangle, wrap well, and chill in the refrigerator for at least 3 hours.
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On a lightly floured work surface, use a floured rolling pin to roll out the dough to a rectangle roughly 8" x 15" in size and about 1/8" thick, with the short side facing you. Fold the dough like a business letter, bringing the bottom third of the dough up and the top third of the dough down and over the top; this is called a letter fold. This is your first turn.
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Rotate the dough 90° (a quarter turn), then repeat the rolling and folding process. As you work, dust the work surface, your hands, and the rolling pin with flour as necessary to prevent sticking. If the dough starts to soften at any point, refrigerate it briefly and then continue.
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Wrap the dough and refrigerate it for at least 1 hour and up to 3 hours.
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Repeat the rolling and folding process one more time (for a total of 3 turns), then wrap the dough well and refrigerate it overnight, or at least 8 hours.
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To shape the bigotes: Line a baking sheeet with a nonstick baking mat or lightly spray with nonstick spray, then line with parchment. On a lightly floured surface, roll the dough into a rectangle that’s slightly larger than 12" x 9" and about 1/4" thick, with the long side facing you. Trim the edges with a pizza cutter or sharp chef’s knife. Cut the dough crosswise into 4 rectangles that are about 3" x 9" in size. Cut each rectangle diagonally, from top left corner to bottom right corner, to make 8 triangles.
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Working with one triangular piece of dough at a time, roll it with a pin or stretch it with your hands to lengthen it to 11". Make a 1/2" cut at the center of the wide end, then roll up the dough towards the point, not too tightly. Use your hands to gently stretch and curve the ends to resemble a mustache. Place it on the baking sheet with the point side down so that it won’t unfurl during baking. Shape the rest of the bigotes.
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Cover the baking sheet with plastic wrap or a reusable cover (like a large, overturned baking dish) and let the bigotes rise in a warm place until they are puffy and nearly doubled, 1 1/2 to 2 hours.
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Towards the end of the rise time, preheat the oven to 400°F with a rack in the center.
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Bake the bigotes for 5 minutes at 400°F, then reduce the oven to 350°F and continue to bake for 15 to 20 minutes, rotating halfway through, until golden brown on the bottom. The bigotes should feel light, but they won’t be shiny like croissants since there’s no egg wash.
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To coat the bigotes: Brush some of the melted butter over all of the bigotes and allow them to sit for 3 to 5 minutes. Put the sugar in a shallow bowl. One at a time, toss a bigote gently in the bowl to cover generously with sugar before serving.
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Storage information: Bigotes are best fresh. Store airtight at room temperature for 1 day.
