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In a mixer bowl, combine strong flour, cake flour, sugar, salt, and yeast. Then add cold eggs and cold water. Mix with a dough hook until the ingredients come together to form a shaggy dough.
Add lukewarm butter to the dough and knead until the butter is fully absorbed and the dough becomes smooth.
Cover the dough with a cotton cloth and let it rest at room temperature for 5 minutes. This helps the gluten to develop.
After 5 minutes, knead the dough again until it is smooth and elastic, stretching thin like bubblegum when tested (windowpane test).
Place the dough in a bowl, cover it, and let it ferment at 27ยฐC for about 120 minutes, or until it has doubled in size.
Once fermented, degas the dough on a lightly floured surface and divide it into 4 equal pieces.
Round each piece of dough, cover with a cloth, and let them rest at room temperature for 10 minutes.
Using a rolling pin, flatten each dough ball into a disc with a thickness of about 1 cm.
Place the flattened dough discs on a baking sheet, cover with a cloth, and let them rest for another 10 minutes.
Dip your fingers into the rested dough, pressing down to the bottom to create several holes. Place a cube of cold butter (about 4-5g each) into each hole and press firmly.
Sprinkle about 20g of sugar evenly over each dough.
Let the dough undergo its final fermentation at 27ยฐC for about 100 minutes, or until it doubles in volume.
Preheat the oven to 190ยฐC. Then, bake the brioche at 170ยฐC for 12-14 minutes, or until golden brown.