In a large bowl whisk together the active sourdough starter, warm milk, sugar, and egg until smooth.
Add bread flour, all-purpose flour, and salt to the bowl and mix with a dough scraper until a rough shaggy dough forms.
Knead the dough briefly by hand until it comes together, then add the softened butter and continue kneading 3–4 minutes until smooth but still slightly tacky.
Shape the dough into a rectangle, cover the bowl tightly, and let rest at room temperature for 30–45 minutes.
Transfer the covered dough to the refrigerator and chill for at least 6 hours or overnight until firm.
Place the cold laminating butter between two sheets of parchment and pound and roll it into a 15 x 15 cm square, then chill until just firm but pliable.
On a lightly floured surface roll the cold dough into a 25 x 25 cm square, place the butter square in the center at a 45° angle, and fold the dough corners over to fully encase the butter.
Roll the butter-encased dough into a long rectangle about 20 x 60 cm, keeping edges straight and thickness even.
Fold the rectangle into thirds like a letter, wrap in plastic, and chill for 30–45 minutes.
Rotate the dough 90°, roll again to 20 x 60 cm, fold into thirds, wrap, and chill for another 30–45 minutes.
Repeat the rolling and folding one more time for a total of three folds, then wrap well and chill for at least 2 hours or overnight.
When ready to shape, roll the laminated dough into a rectangle about 25 x 40 cm, trimming edges to neaten if necessary.
Using a ruler and sharp knife or pizza cutter, cut the dough into long triangles with a base of about 8–9 cm and full height of the rectangle.
Place 2–3 chocolate batons or a tablespoon of chopped chocolate along the base of each triangle, pressing lightly so they adhere.
Starting at the base, roll each triangle snugly toward the tip without squashing the layers, slightly stretching the tip as you roll.
Arrange the rolled croissants on parchment-lined baking sheets with the pointed tips tucked underneath and spaced well apart.
Cover the trays loosely with plastic wrap or a large bag and proof at a warm room temperature until puffy, jiggly, and nearly doubled, about 4–6 hours depending on starter strength and room temperature.
Near the end of proofing, whisk together the egg and milk for the egg wash until smooth.
Preheat the oven to 200°C (390°F) with racks in the upper and lower thirds.
Gently brush the croissants with a thin, even layer of egg wash, avoiding the cut edges to preserve lamination.
Bake the croissants for 10 minutes at 200°C (390°F), then reduce heat to 180°C (355°F) and bake another 8–10 minutes until deep golden brown.
Rotate the trays halfway through baking for even browning if needed.
Transfer baked croissants to a wire rack and let cool at least 20 minutes before serving so the crumb sets and chocolate thickens slightly.