Combine 1 cup flour, 2 tablespoons brown sugar, 1 teaspoon salt, and instant yeast in a mixing bowl. Add the water and room temperature butter and beat for 1 minute. Stir in the remaining cup of flour and mix well. Turn the dough out onto a lightly floured surface and knead until smooth and elastic, about 10 minutes. Shape the 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.

Make the filling: Place chopped apples and 3 tablespoons butter in a medium saucepan. Combine brown sugar, flour, salt, cinnamon, and nutmeg; add the mixture to the saucepan. Bring to a boil over medium-high heat, then simmer until apples are tender, approximately 5 minutes. Cool filling before use.

Punch the risen dough down and let rest, covered, for 15 minutes. Roll the dough into a 13 x 8-inch rectangle. Transfer to a greased or parchment-lined baking sheet.

Spread apple filling down the center third of the dough. Cut and braid the dough to resemble a long braid. Pinch both ends to seal filling. Cover and let rise for 30 to 40 minutes until puffy.

Bake the Danish at 375°F (190°C) for 20 minutes. Cover with foil for the last 10 minutes to prevent over-browning. Cool completely.