Mexican Concha (Sweet Cookie-Topped Buns)

Directions

1. For Alice Medrich's brioche:
2. Spread the flour in a shallow baking pan. Cover and refrigerate or freeze until needed, at least 45 minutes.
3. Use the paddle attachment to whip the cold butter only until it is smooth and slightly fluffy, but still cold. There should be no small hard lumps when you pinch it between your fingers. Scrape the butter into a mound on a piece of wax paper and refrigerate it until needed. Wash the mixer bowl. (Proceed with the recipe right away; too long a delay will re-harden the butter.)
4. Dissolve the yeast and 1 teaspoon of sugar in the water in the bottom of the clean mixer bowl, now fitted with the dough hook. Add the remaining 1/3 cup (66 grams) sugar, eggs, yogurt, salt, and flour. Mix on low speed until the ingredients are blended, scraping the bowl as necessary. Turn the speed up to medium (speed 5) and knead for 5 minutes. At the end of the kneading period the dough will be soft, moist, and sticky, and very elastic. All or most of it will be wrapped around the dough hook. Add the cold creamed butter in several large pieces, pushing it into the dough, and beat with the hook until it is well blended. You will need to stop the mixer several times to scrape the dough from the sides of the bowl and pull it completely off of the hook, until the butter appears thoroughly incorporated. Scrape the dough into a bowl, cover, and refrigerate overnight, or up to 24 hours.
5. Scrape the cold dough out onto a well-floured surface. Divide it into 18 to 20 balls, then shape each one into a rounded bun, using the heel of your hand to drag the lump over your work surface so that you have a smooth ball. Place each ball on parchment-lined baking sheets, leaving at least two inches between each one.
6. Cover the pan loosely with plastic wrap and let rise in a warm place until almost doubled, about 2 hours. Now make the cookie dough.

1. For Rick Bayless' cookie tops and shaping:
2. While the dough doing its second rise, add the butter and sugar to the bowl of the stand mixer fitted with a paddle attachment. Mix at medium speed until thoroughly combined.
3. Reduce the speed to low, then slowly add the flour, cinnamon (or matcha), and salt until the mixture is smooth. Transfer half the cookie dough to a sheet of parchment paper, cover with another sheet of parchment paper, and roll out to 1/4- to 1/8-inch thick. Slide onto a baking sheet and refrigerate for at least 1 hour. Repeat with the other half of the dough.
4. Heat the oven to 350° F. Use a round cookie- or biscuit-cutter to cut out circles of the dough that will cover the balls. When ready to bake, beat the egg with 1 tablespoon of water, brush the concha with the egg wash, and lay a cookie topping over each concha. Press it lightly onto the surface. With a sharp knife, score the dough: The traditional shell pattern is a series of curved lines, like a seashell.
5. Bake for 25 to 30 minutes, rotating the baking sheet halfway through.