Mix 25g butter and 35g sugar by hand with a whisk until creamy white, about 3 minutes. Whisk the egg, weigh it, and measure out 1/4 of the weight. Add the egg quarter to the butter mixture and stir until smooth, about 2 minutes. Sieve 80g flour with 1/4 teaspoon baking powder. Add the flour mixture one spoonful at a time to the butter mixture, kneading by hand after each addition until the dough is even. Wrap tightly in cling film and refrigerate for at least 1 hour.
Mix flour, sugar, salt, and dry yeast in a large bowl. Warm milk to lukewarm (35°C / 95°F), mix with the remaining egg, and pour into the flour mixture. Stir roughly with a wooden spoon until combined. Turn onto a worktop and knead by repeatedly lifting and slapping the dough down hard for 10–15 minutes until no longer sticky. Distribute butter over the dough and knead, roll, and slap for another 10 minutes until smooth. Divide into 5 equal portions (use a scale), shape each into a ball, and let rise in a warm place for 40 minutes.
Divide the chilled shortcrust pastry into 5 equal portions using a scale. Shape each portion into a ball, then press flat between two plastic sheets until you have 5 thin cakes about the size of your palm.
Place a yeast dough ball on top of each shortcrust flatbread. Wrap the shortcrust tightly around the yeast ball and seal. Roll the smooth side of each melon pan in sugar to coat. Using a sharp knife, press three parallel cuts into the top. Rotate the melon pan 45 degrees and press three more parallel cuts to create a cross-hatch pattern.
Place the melon pans on a baking sheet lined with baking paper. Bake in a preheated oven at 170°C (338°F) for 12 minutes until light golden brown.
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