Step 1

Stir together milk, yeast, and 1 teaspoon of the sugar in a small bowl. Let stand until foamy, about 5 minutes.

Step 2

Beat flour, salt, and remaining 3 tablespoons sugar in a stand mixer fitted with a dough hook on medium-low speed until combined, about 30 seconds. Add milk mixture and egg; beat on low speed until almost fully combined, about 3 minutes. Add butter; beat until dough is smooth and elastic, 4 to 5 minutes. Scrape down sides of bowl. Shape dough into a neat ball using your hands. Cover bowl with plastic wrap. Let stand at room temperature until dough doubles in size, about 1 hour.

Step 3

Grease a 9-inch square baking pan. Sprinkle ginger over bottom of pan; set aside. Melt butter in a 12-inch skillet over medium-high. Sprinkle sugar and salt evenly over butter; cook, stirring occasionally, until no clumps of sugar remain, about 1 minute. Arrange apples, rounded side down, in skillet. Reduce heat to medium; cook, swirling skillet occasionally, until caramel bubbles vigorously, thickens, and turns a deep amber color, 10 to 15 minutes. You may have to rotate the skillet or stir very carefully to ensure caramel browns evenly and apples cook properly. The apples should be browned in spots, but they don’t need to be completely cooked at this point. Remove from heat. Carefully arrange apples in prepared baking pan, rounded side down. Pour caramel over apples. Set aside until ready to use.

Step 4

Beat almond paste, butter, brown sugar, and salt with an electric mixer on medium speed until smooth and creamy, about 1 minute. Set aside until ready to use.

Step 5

Turn dough out onto a clean work surface; shape into a 16- x 9-inch rectangle, with the long edge parallel to work surface. Spread Filling evenly over right half of dough. Fold the left half over filling; pinch edges to seal. Place filled dough inside baking pan over Topping. Cover loosely with plastic wrap. Let stand at room temperature until dough almost doubles in size, about 30 minutes. Meanwhile, preheat oven to 375°F.

Step 6

Uncover dough; place baking pan on a rimmed baking sheet lined with aluminum foil. Bake in preheated oven until bread is deep golden brown, puffed, and sounds hollow when you knock it in the center, about 25 minutes. Let cool on a wire rack 5 to 10 minutes. Carefully run a knife around edges of baking pan to loosen bread; invert bread onto a platter. Serve with crème fraîche or whipped cream and strong coffee.