Prepare the dough. In a cup or bowl, heat the milk in the microwave in 15-second increments or on the stove until it's just warm to the touch. A thermometer should read around 110°F-115°F.
Stir in the yeast and 1 tablespoon of the sugar. Loosely cover with plastic wrap and let it stand for 5 minutes until it gets very foamy. If the mixture doesn't get foamy, throw it out and start again with a new packet of yeast.
In the bowl of a stand mixer (or a regular large bowl), whisk together the all-purpose flour, remaining sugar, and salt.
Attach the dough hook to the mixer and add the melted butter, eggs, and yeast mixture. Mix on medium-low speed for 6-8 minutes (or mix by hand), until the dough becomes smooth and elastic. The dough will be slightly tacky but not completely sticky. If it is still fairly sticky, knead in 1 tablespoon of additional flour at a time until smooth and elastic.
Coat a large bowl with oil and transfer the dough to the bowl, shaping it into a ball. Cover it with a kitchen towel and let it rise for 2 hours until doubled in size.
Line 2 large baking sheets with parchment paper and set aside.
Divide the dough into 12 equal pieces and shape into balls. Place 6 balls on each of the prepared baking sheets and cover loosely with clean kitchen towels. Let the balls rise again for 1.5 hours in a warm place, or until doubled in size.
Prepare the topping. Beat together the flour, powdered sugar, and butter in a large mixing bowl with a handheld mixer until light and fully.
Divide the mixture into 2 equal pieces and place each halve in a small bowl.
Make the white vanilla topping by adding vanilla extract to half of the dough, and beat until combined. Gather the dough with your hands, divide it into 6 equal pieces and shape into balls.
Make the brown chocolate topping by adding cocoa powder to the other half of the dough, and beat until combined. Gather the dough with your hands, divide it into 6 equal pieces, and shape into balls.
Lightly flatten each ball in between two sheets of plastic wrap using a tortilla press. You could also use a rolling pin to shape the topping into rounds.
Assemble the conchas. Carefully remove the rolled out topping from the plastic wrap and drape it onto the risen concha dough. Lightly pat it down to secure the topping onto the dough.
Preheat oven to 350°F. While the oven is preheating, cut grooves into the topping using a paring knife so it resembles a clam shell, a criss cross pattern, or whichever pattern you prefer.
Bake for 20-25 minutes, until the dough is lightly golden brown. Allow to cool for 10 minutes before serving.
Notes
Make sure your yeast is alive before adding it to the dough. If the milk and yeast mixture is bubbly and frothy, that means the yeast is alive and thriving which is great! If it's not, that means the yeast is likely dead and you need to start over with a new yeast packet.
To make shaping the topping a little easier, you could use concha shell cutter.
Storage: Store leftover conchas in an airtight container in a cool and dry place for up to 5 days.