In a medium liquid measuring cup, combine the warm milk, yeast, and 1 tablespoon of sugar. Let it stand for 5 to 10 minutes until the yeast creates foam. Add the cold water.

In the bowl of a stand mixer fitted with the dough hook, whisk together the flour, remaining sugar, and salt. Add the yeast mixture and very soft butter. Mix on low speed for 1 minute until shaggy but combined.

Scrape down the sides of the bowl and continue to mix on low for 2 more minutes. The dough should be soft but not sticky.

Remove the dough from the bowl and shape it into a square about ½ inch thick. Wrap it tightly in plastic wrap and refrigerate for 30 minutes.

Soften the butter for the layer slightly at room temperature for 15 minutes. Cut two sheets of parchment paper to 12-inch squares and shape the butter into a 10-inch square.

Refrigerate the butter for about 10 minutes.

Roll the chilled dough into a rectangle, around 10×20 inches and about ¼ inch thick.

Place the butter in the center of the dough and fold the dough around the butter, sealing it in.

Roll the dough into a 10×18 inch rectangle. Trim the edges and perform a letter fold by folding the dough into thirds.

Cover with plastic wrap and refrigerate for 30 minutes. Repeat this rolling and folding process three more times, refrigerating after each turn.

Wrap the dough tightly and refrigerate overnight or up to 18 hours.

On the next day, roll the dough into a 12×28 inch rectangle. Cut the triangles for croissants by marking the dough in 4-inch increments.

Shape each triangle by making a cut at the base, folding corners inward, and rolling the triangle up towards the tip.

Place the shaped croissants on baking sheets, cover loosely, and let rise at room temperature for 2½ to 3 hours until doubled in size.

Preheat the oven to 425°F. Whisk together the egg and heavy cream for an egg wash.

Brush the tops of the croissants with the egg wash and bake for 15 to 20 minutes at 375°F until golden brown.

Let cool on the baking sheets for about 10 minutes before serving.