Weigh the butter and let it soften; bring the egg to room temperature. Sift together the cake flour, almond flour, and salt.
90 g unsalted butter, 1 whole egg, 220 g cake flour, 40 g almond flour, 2 pinches fine Guérande salt
Add the sugar to the butter and mix with a spatula until smooth and pliable. Beat the egg, then add it to the butter in 4 to 5 additions, emulsifying well after each addition.
100 g superfine sugar
Add the sifted dry ingredients and mix with a spatula. Once partly incorporated, mix by hand until no streaks of flour remain, then knead briefly about 10 times.
Divide the cookie dough into 2 portions, wrap, and flatten. Refrigerate the portion you will use right away and freeze the other; let rest for at least 1 hour (ideally 6 to 8 hours). If needed, thaw in the refrigerator 1 to 2 hours before using.
Preferment
Warm the milk to about 30 °C, sprinkle the yeast over it, and let stand. In a bowl, mix the bread flour and sugar, then add the milk-yeast mixture and stir until smooth.
140 g bread flour, 15 g cane sugar, 95 g whole milk, 1.5 g instant dry yeast
Knead lightly about 1 minute to eliminate dry spots, shape into a ball, then ferment at 30 °C about 1 hour, until ~1.8 times the original volume.
Final mix
Weigh the final-mix ingredients and let the butter soften. Tear the preferment into ~10 pieces.
20 g bread flour, 40 g all-purpose flour, 10 g cane sugar, 10 g buttermilk powder, 3.5 g salt, 1 egg yolk, 20 g milk, 20 g unsalted butter
Combine the flours, sugar, salt, and buttermilk powder. Mix the egg yolk and milk, then add to the dry ingredients and mix partially. Add the pieces of preferment and knead; when the dough is ~80% incorporated, add the butter and knead until smooth.
First fermentation: Shape into a ball, place in a bowl, and cover. Ferment at 30 °C about 1 hour, until doubled (finger test).
Divide the dough into 6 portions, shape into balls, cover, and rest 15 minutes.
Cookie dough (portioning)
During the first fermentation, divide the cookie dough into 6 portions; knead each portion 4 to 5 times and shape into balls. Keep well chilled in the refrigerator if made ahead. For the chocolate version, fold the chocolate chips into the cookie dough.
5 g chocolate chips
Shaping and finishing
Flatten each portion of cookie dough into a disk slightly larger than the bread dough ball. Round the bread dough again, then wrap it with the cookie dough; flip so the cookie side is up and round.
Sprinkle the topping sugar evenly over the cookie surface and score a light crosshatch without cutting too deeply.
30 g granulated sugar
Second proof and baking
Proof at 30 °C for 45 minutes (no steam). Preheat the oven to 200 °C so it’s ready at the end of proofing.
Lower the oven to 190 °C and bake 13 to 15 minutes, until lightly colored. Cool on a rack.
Notes
For the best contrast with the exceptionally soft, airy crumb, make the melon pan fairly large.
The cookie dough quantity is enough for 2 batches, so you do not have to use a partial egg; for a single batch, halve all the cookie dough ingredients.
For the topping, regular granulated sugar works better than very fine sugar.
If the cookie dough rests only briefly, the surface will be crispier and more likely to crack during baking.