Warm the milk to 105–110°F. Sprinkle in the active dry yeast and a pinch of sugar, then whisk to combine. Let stand for about 5 minutes, or until foamy.

In the bowl of a stand mixer fitted with the paddle attachment, mix the eggs, activated yeast mixture, and vanilla extract on low speed just until combined. With the mixer running, gradually add the ½ cup sugar and kosher salt, mixing until incorporated.

Knead on low speed for about 2 minutes, until the dough comes together and begins pulling away from the sides of the bowl.

Continue kneading for about 5 minutes, until the dough comes back together and becomes more cohesive and elastic. It should remain slightly tacky, as we'll finish kneading it by hand.

Stop adding flour once you reach the desired consistency. Continue kneading for 5–8 minutes, until the dough holds its shape while remaining slightly tacky.

Shape the dough into a smooth ball by tucking the edges underneath as you rotate it, applying gentle pressure to tighten it. Place in a large bowl greased with softened butter, turning to coat evenly.

Using the paddle attachment, beat the shortening and ½ cup sugar on medium-high speed for 3–5 minutes, until creamy, lighter in color, and slightly fluffy. Scrape down the bowl as needed.

Add the vanilla bean paste and 1 cup of flour. Mix on the Stir setting to incorporate, then increase to medium-low speed until a smooth, pliable paste forms.

Divide the topping evenly in half. Add the cocoa powder to one half and mix briefly with the paddle attachment until fully incorporated.

Divide the vanilla topping into 5 equal portions and the chocolate topping into 6 equal portions. Roll each into a ball, flatten slightly, and set aside.

Transfer the risen dough to a clean, nonstick work surface, lightly floured only if needed to prevent sticking. Avoid excess flour, which can dry it out. Divide into 3 oz portions.

Shape each portion into a smooth, round ball by tucking the edges underneath as you rotate, gently tightening the dough. For a smoother finish, cup your hand over it and roll in a circular motion. Lightly coat them with softened butter.

Lightly butter your hands and flatten a portion of sugar topping into a circle large enough to cover the surface of the dough ball.

Gently press the topping into place, smoothing it over the top and partway down the sides while slightly flattening the dough into a thick, round disk. Repeat with the remaining portions.

Arrange on two parchment-lined half-sheet pans, leaving enough space to expand during proofing and baking.

Lightly flour a concha stamp and press firmly into the sugar topping, scoring only the topping without cutting into the dough. Alternatively, use a knife to score the design.

Loosely cover the conchas with plastic wrap, keeping it from pressing against or sticking to the topping. Place in a turned-off oven with the light on and the door closed. Let rise for 1½–2 hours, or until noticeably puffy and nearly doubled in size.

When the conchas are nearly ready, remove them from the oven and preheat to 350°F. I removed mine after 1 hour and 20 minutes, allowing them to finish rising at room temperature while my oven preheated for about 10 minutes.

Carefully remove the plastic wrap and bake for 15–20 minutes, rotating the pans halfway through for even baking.

The conchas are ready when the bottoms are lightly golden brown and the bread is baked through, with an internal temperature of 205°F. Check each pan individually, as one may finish sooner.

Let the conchas rest on the baking sheets for 5 minutes, then carefully transfer them to a wire rack to finish cooling. Enjoy warm or at room temperature.