Add the active sourdough starter, flour, sugar, salt, cooled melted butter, milk, and 120 g water to the bowl of a stand mixer.
Mix with the dough hook until the ingredients come together into a firm dough. If the dough feels too dry, add 1 tablespoon of water at a time, only as needed.
Knead until the dough is smooth, evenly mixed, and slightly firm. The dough should not feel sticky or loose.
Cover the dough and let it rise at room temperature for a few hours, just until slightly puffed. Do not let it fully double.
Flatten the dough into a square or rectangle, wrap it tightly, and refrigerate overnight.
Make the butter block by placing the cold butter between two sheets of parchment paper. Pound and roll it into an even 8x8-inch square. Refrigerate until firm but still pliable.
The next day, remove the butter block from the fridge a few minutes before the dough. The butter should be cold but bend slightly without cracking.
Roll the chilled dough into a 12x12 square.
Place the butter block in the center of the dough, fold the dough over it, and pinch the edges closed to seal the butter inside.
Roll the dough into a long rectangle, about 10x20 inches, working gently so the butter stays enclosed and the dough does not tear.
Fold the dough into thirds like a letter. Wrap and refrigerate for 30 to 60 minutes.
Rotate the dough 90 degrees, roll it out again into a 10x20 rectangle, then complete a second letter fold. Wrap and refrigerate for another 30 to 60 minutes.
Rotate the dough again, roll it out one more time, then complete a third letter fold.
Wrap the dough and refrigerate overnight, or for at least several hours, so the dough can relax before shaping.
Roll the chilled dough to about ¼ inch thick, 12x24 inches long.
Cut the dough into 12 equal triangles, using a ruler if needed. Trim the edges first, then mark even sections and cut diagonally to create long triangles. Make a small slit in the wide end of each triangle, gently stretch the dough, then roll from the wide end toward the point.
Place the shaped croissants on parchment-lined baking sheets with plenty of space between each one.
Cover loosely and let the croissants proof until very puffy, airy, and slightly wobbly. This can take 8 to 12 hours or longer, especially in a cool kitchen. Watch the croissants, not the clock.
Preheat the oven to 425°F.
Whisk together the egg yolks and milk for the egg wash. Gently brush the tops of the croissants without pressing them down.
Bake at 425°F for 10 minutes, then reduce the oven temperature to 400°F and bake for another 10 to 15 minutes, or until the croissants are deeply golden brown.
Let the croissants cool slightly on the baking sheet before serving.