In a large bowl, combine the all-purpose flour, bread flour, sugar, salt, and instant yeast. Whisk them together.

Add the cold, cubed butter into the dry mixture. Work the butter into the flour until crumbly.

In a separate bowl, whisk the warmed milk with the sourdough starter and cold water until smooth. Gradually add this to the dry ingredients while stirring.

Transfer the dough to a lightly floured surface and knead for 5-7 minutes until smooth and elastic.

Shape the dough into a ball and place it in a lightly oiled bowl. Cover and let it rest for 1-2 hours until doubled.

Gently punch the dough down and roll it out into a rectangle about 1/4 inch thick. Fold it like a letter in thirds and refrigerate for 30 minutes.

Repeat the rolling and folding twice more, refrigerating for 30 minutes in between each session.

After the final roll, cut triangular shapes from the dough. Roll each triangle from base to tip to form a crescent shape.

Place each croissant on a baking sheet. Cover and let them rise for another 1-2 hours until puffy.

Preheat the oven to 375°F (190°C). Brush each croissant with beaten egg. Bake for 15-20 minutes until golden brown.

Allow the croissants to cool on a wire rack. Serve warm.