Cronuts

• Place yeast into the mixing bowl of a large stand mixer. Whisk in warm water and let stand until a creamy foam forms on top, about 5 minutes. Add salt, sugar, milk, 2 tablespoons melted butter, vanilla extract, egg, and nutmeg. Whisk mixture thoroughly. Pour flour on top of liquid ingredients. Place mixing bowl onto mixer.
• Attach dough hook to mixer and knead on low speed until dough comes together in a ball and becomes soft and sticky, about 3 minutes. Dough will stick to the hook and pull away from the side of the bowl.
• Transfer dough to a floured work surface, knead 2 or 3 times, and shape into a ball. Wrap dough in plastic and refrigerate for 20 minutes to let gluten relax.
• Remove dough from refrigerator, unwrap, and dust lightly with flour. Roll out into a 9×18-inch rectangle about 1/4-inch thick. Evenly spread 1/2 of the softened butter onto the middle third of the dough. Fold one unbuttered third over the buttered third and press lightly; spread remaining butter on top of that third. Fold remaining third over the first (buttered) third. Transfer dough onto a sheet pan, cover lightly with plastic wrap and a kitchen towel, and refrigerate 20 to 30 more minutes for butter to become firm. Sprinkle dough lightly with flour as you work if it becomes sticky.
• Return dough to floured work surface and pat very gently into an 8×14-inch rectangle about 1/2-inch thick. Fold outer thirds over center third as before; roll out into an 8×14-inch rectangle again, keeping the edges as straight as possible. Fold in thirds as before. Cover dough lightly with a kitchen towel, and refrigerate for 2 hours.
• Roll the dough out to about 3/8-inch thick. Cut dough in half crosswise. Leave half on a lightly floured work surface; refrigerate other half of dough until needed.
• Use a sharp 3-inch round cutter to cut 8 circles of dough from the piece on the work surface. Use a 1-inch round cutter to cut the donut holes out of the dough circles.
• Line a baking sheet with waxed paper and sprinkle lightly with flour. Arrange cronuts and holes onto the prepared baking sheet. Let rise in a draft-free, warm place (such as an unheated oven) until doubled in size, about 1 hour.
• Heat vegetable oil in a deep fryer or a deep saucepan over medium heat to 350 degrees F (175 degrees C).
• Carefully lift a risen cronut from the baking sheet and gently drop it into the hot oil. Fry two at a time until golden brown, 1 1/2 to 2 minutes per side. Cronuts will puff up as they cook. Drain on racks over paper towels; let cool. Repeat to fry cronut holes.

Glaze

• Whisk confectioners' sugar, milk, and vanilla for glaze in a shallow bowl until smooth and slightly runny.

Finishing

• Pick up a fried cronut and gently dip the top in glaze; return to the rack and let stand until glaze has set, about 15 minutes.
• Alternately coat some cronuts with cinnamon sugar