Place the milk and 1 tablespoon of the granulated sugar in a microwave-safe bowl. Heat in 15–20-second intervals, stirring between each, until the sugar dissolves and the milk reaches 110°F / 43°C. Do not overheat. Stir in the yeast and let it bloom until frothy and bubbly, about 10 minutes.
In the bowl of a stand mixer fitted with the whisk attachment, combine the bread flour, remaining sugar, and salt until evenly mixed. Switch to the dough hook attachment.
In a small bowl, whisk together the vanilla and eggs until combined. With the mixer on low speed, slowly add the yeast mixture to the flour, followed by the eggs.
Mix until well combined, 3 to 5 minutes. Increase the speed to medium and begin adding the softened butter, one piece at a time, allowing each to incorporate before adding the next. About 4 to 6 minutes total. Scrape down the bowl and dough hook as needed.
Continue kneading until the dough becomes smooth, elastic, and pulls cleanly away from the sides of the bowl, about 10 minutes.
Transfer the dough to a lightly greased large bowl. Cover with greased plastic wrap and let rise until doubled in volume, about 1 hour.
While the dough rises, prepare the topping. Sift together the powdered sugar and bread flour. In a separate bowl, combine the vegetable shortening, vanilla, and the sifted dry ingredients.
Using clean hands, mix until homogeneous with a playdough-like consistency.
Lightly dust the work surface with flour and knead the topping dough briefly. Divide into 12 portions, roll into balls, and keep covered until ready to use. Line two rimmed baking sheets with parchment paper and set aside.
After the dough has doubled in size, turn it out onto the lightly oiled or floured work surface. Pull the dough toward the center and press it inward. Repeat from all sides to degas and strengthen the dough.
Divide it into 12 equal portions. Cover loosely with greased plastic wrap.
Working with one piece at a time, shape the dough by pulling the edges inward and pinching to seal, creating a smooth top.
Place seam-side down and roll gently against the counter with a cupped hand until round and slightly taut. Repeat with the remaining pieces.
Place 6 dough balls on each prepared baking sheet, spacing them evenly. Press slightly to flatten, forming a small dome. Rub a small amount of vegetable shortening between your palms and lightly coat the top of each dough ball.
Lightly flour your hands. Flatten each topping portion into a thin, round disk large enough to cover the dough ball. Alternatively, place one portion between two pieces of plastic in a tortilla press and press it into a disk slightly larger than the diameter of the dough ball.
Place the topping disk on top and press gently. Using a concha cutter or a knife dusted with flour or powdered sugar, score the shell pattern.
Place the prepared sheets in a warm area and let the conchas rise again until doubled in size, 1 to 1½ hours.
Adjust the oven racks to the upper-middle and lower-middle positions and preheat the oven to 350°F (175°C).
Bake for 20 to 25 minutes, switching and rotating the baking sheets halfway through, until the conchas are golden brown and the internal temperature reaches at least 205°F.
Transfer the baking sheets to wire racks and let cool for 15 minutes before serving.