Make the laminated dough:

1. Combine the warm milk, yeast, and 1 tablespoon of the sugar. Let sit for 5–10 minutes until foamy.
2. In a large bowl, whisk together the flour, salt, and remaining 2 tablespoons sugar. Add the yeast mixture and egg. Mix until a soft dough forms. Knead briefly — just until smooth. Do not over-develop the gluten. Wrap in plastic and refrigerate for 30 minutes.
3. On a lightly floured surface, roll the chilled dough into a 12 x 8-inch rectangle. Place the cold butter between two sheets of plastic wrap and beat with a rolling pin into an 8 x 6-inch rectangle (about 1/4 inch thick). Place the butter rectangle in the center of the dough. Fold the dough sides over the butter to enclose it fully, like a letter. Press the edges to seal.
4. Roll the dough out into a long rectangle (about 18 x 6 inches), then fold it in thirds (like a letter). Wrap and refrigerate for 30 minutes. This is one “turn.” Repeat for a total of 3 turns, chilling 30 minutes between each. After the final turn, refrigerate overnight (or at least 4 hours).

Assemble and bake:

1. Remove dough from the refrigerator and let sit at room temperature for about 15 minutes. Preheat oven to 350°F.
2. Make the filling: mix the softened butter, sugars, and cinnamon together.
3. Roll the dough into an 18 x 10-inch rectangle. Spread the filling down the center third. Slice from each outer edge to the filling at 1/2-inch intervals on both sides. Fold the strips over the filling, alternating sides, to form a crosshatch braid.
4. Transfer to a parchment-lined baking sheet. Bake about 30 minutes until puffed and golden brown.
5. Make the glaze: combine the melted butter, powdered sugar, and vanilla. Add hot water until you reach the desired drizzling consistency. Drizzle over the cooled danish.