Step 1

Combine brown sugar, butter, and miso in a small saucepan. Cook over medium, whisking often, until mixture is smooth, 3 to 4 minutes. Whisk in 1/2 cup heavy cream. Bring to a boil over medium. Boil, whisking constantly, 1 minute. Transfer caramel to a small bowl, and stir in salt. Set aside.

Step 2

Line 2 baking sheets with parchment paper; set aside. Uncover refrigerated croissant dough; transfer to a lightly floured surface. Lightly flour top of dough; roll into a 21- x 11-inch rectangle. Using a pizza cutter, trim edges to form a 20- x 10-inch rectangle; discard scraps. Cut dough into 8 (5-inch) squares. Place squares, slightly overlapping, in an even layer on 1 prepared pan; cover tightly with plastic wrap, and refrigerate 15 minutes.

Step 3

Fold 1 dough square in half diagonally to form a triangle. Using a sharp knife, make a cut parallel to 1 open side of the triangle, starting at the base 1/2 inch from the corner and stopping 1 inch from the top point of the triangle. Repeat cut on second open side, leaving top point of triangle attached. Unfold triangle. Lift the 2 outer border pieces, and cross them over each other, aligning border edges with edges of inner square. Transfer to prepared pan. Repeat with remaining dough squares, placing at least 2 inches apart on both prepared pans.

Step 4

Place 6 to 7 apple slices, slightly overlapping, inside border of 1 pastry. Brush apple slices with 1 teaspoon caramel. (If caramel is too thick, place in a microwavable bowl, and microwave on high 20 seconds to thin.) Repeat with remaining pastries, apple slices, and caramel. Reserve remaining caramel for serving.

Step 5

Whisk together egg and milk in a small bowl. Brush borders of pastries lightly with egg mixture, using a paper towel to wipe away excess. Reserve remaining egg mixture in refrigerator. To proof pastries, add 2 cups hot water to a small bowl, and place in a cold oven. Place baking sheets with pastries, uncovered, in oven with hot water. Close door, and proof until pastries double in size and jiggle slightly when you shake baking sheets, 30 minutes to 1 hour and 30 minutes. Remove pastries and water from oven; let pastries stand at room temperature 30 minutes. Preheat oven to 425°F. Place oven rack in middle position.

Step 6

Lightly brush borders of pastries on 1 baking sheet with a second coat of egg mixture. Place baking sheet on middle rack in preheated oven, and immediately decrease temperature to 375°F. Bake until golden brown and crisp, 20 to 26 minutes. Transfer pastries to a wire rack. Return oven temperature to 425°F. Repeat process with egg mixture and remaining pastries. Serve warm, or let cool completely, about 1 hour.

Step 7

Combine sour cream and remaining 1/2 cup heavy cream in bowl of a stand mixer fitted with the whisk attachment. Beat on medium-high speed until soft peaks form, 1 to 2 minutes. Dollop cream mixture onto pastries, and drizzle each with 1 tablespoon caramel. Serve remaining caramel on the side.