To make the pastry: Combine 1 cup flour, 2 tablespoons brown sugar, 1 teaspoon salt, and instant yeast in a mixing bowl. Add water and room temperature butter; beat for 1 minute. Stir in remaining cup of flour and mix well. Turn dough out onto a lightly floured surface and knead until smooth and elastic, about 10 minutes. Shape dough into a ball and transfer to an oiled bowl, turning to coat the surface of the dough. Cover the bowl with plastic wrap and let the dough rise until doubled in bulk, about 1 hour.

Meanwhile, make the filling: Place chopped apples and 3 tablespoons butter in a medium saucepan. Combine 1/2 cup brown sugar, 1 ½ tablespoons flour, 1/4 teaspoon salt, cinnamon, and nutmeg and mix well; add mixture to the saucepan; bring to a boil over medium-high heat and then reduce heat to a simmer. Cook until apples are tender, about 5 minutes. Remove from heat and allow filling to cool.

Transfer Danish pastry dough to a floured surface and punch down. Let dough rest, covered, for 15 minutes. Roll dough out into a 13 x 8-inch rectangle. Place dough on a greased or parchment-lined baking sheet.

Turn the baking sheet so that you face the short side of the dough. Spread apple filling down the center third of the dough. With a sharp paring knife, make cuts in the dough along the right side, starting each cut about 1/4 inch from the apple filling. Cut to the edge of the dough strip, with each cut angled to about 4 o'clock. Each strip of dough should be about 1-inch thick. Repeat on the left side of the dough, angling the cuts to 8 o'clock.

Starting at the top, fold the dough strips across the apple filling, alternating left and right. The pastry will look like a long braid. Pinch the top and bottom ends of the braid to seal in the filling. Cover the pastry and let it rise at room temperature until the dough is puffy, 30 to 40 minutes.

Preheat an oven to 375 degrees F (190 degrees C). Bake the braid for 20 minutes, covering it with aluminum foil during the last 10 minutes of baking to prevent over-browning. Remove the pastry from the oven and allow it to cool.

To make the glaze: Heat 2 tablespoons of butter in a saucepan over medium heat. Swirl the pan occasionally and cook until the milk solids turn brown and give off a nutty fragrance, about 5 minutes. Remove from heat and transfer to a bowl to cool slightly. Stir in confectioners' sugar and vanilla. Mix in milk a teaspoon at a time until glaze is thin enough to drizzle. Decorate the Danish braid with glaze.