In the bowl of a stand mixer, whisk together the warm milk, sourdough discard, sugar, and instant yeast until mostly smooth.
Add the flour, salt, and melted cooled butter to the bowl and mix with a spoon until a rough dough forms.
Knead with the dough hook on medium-low speed (or by hand) for 5–7 minutes until the dough is smooth, slightly tacky, and elastic.
Shape the dough into a rectangle, cover the bowl tightly with plastic wrap, and refrigerate for at least 4 hours or overnight to chill completely.
Place the 250 g cold butter between two sheets of parchment paper and pound with a rolling pin into a thin rectangle about 7 x 5 inches (18 x 13 cm).
Square off the edges of the butter block with the bench scraper, chill it in the refrigerator until firm but pliable, about 15–20 minutes.
On a lightly floured surface, roll the chilled dough into a rectangle about 14 x 10 inches (35 x 25 cm), keeping edges as neat as possible.
Place the butter block in the center of the dough so the short sides of the butter run parallel to the long sides of the dough.
Fold the two short ends of the dough over the butter to meet in the middle, then pinch seams and edges well to fully encase the butter.
Rotate the dough 90 degrees so the long seam runs left to right, then roll gently into a long rectangle about 20 x 10 inches (50 x 25 cm).
Fold the top third of the dough down to the center and the bottom third up over it to make a letter fold, then wrap in plastic and chill for 30–45 minutes.
Remove the dough, place it with the open ends facing you, and roll again into a 20 x 10 inch (50 x 25 cm) rectangle.
Repeat the letter fold, wrap in plastic, and chill for another 30–45 minutes.
Perform a final roll to 20 x 10 inches (50 x 25 cm), fold into a letter again, wrap tightly, and chill for at least 1 hour or up to overnight.
When ready to shape, roll the dough on a lightly floured surface into a large rectangle about 24 x 12 inches (60 x 30 cm), about 1/4 inch (6 mm) thick.
Trim the edges with a sharp knife to square them, then mark a line along the long side and cut long, tall triangles about 4 inches (10 cm) wide at the base.
Make a small 1/2 inch (1 cm) slit in the center of each triangle base, gently stretch the tip, and roll each triangle from base to tip, slightly pulling as you roll.
Curve the ends of the rolled croissants inwards to form a crescent shape and place them on parchment-lined baking sheets, spaced well apart.
Cover the croissants lightly with plastic wrap or a clean kitchen towel and proof at room temperature until puffy, jiggly, and nearly doubled, 2–3 hours depending on temperature.
About 20 minutes before baking, preheat the oven to 400°F (200°C) with racks in the upper and lower thirds.
Whisk together the egg and 15 g milk, then gently brush the risen croissants with a thin, even layer of egg wash, avoiding pooling around the base.
Bake the croissants for 18–22 minutes, rotating sheets halfway, until deeply golden brown with visible layers and crisp edges.
Transfer croissants immediately to a wire rack to cool slightly before serving warm or at room temperature.