Heat the oven to 375°F and line a large baking sheet with parchment paper.

Melt the butter in a skillet over medium heat. Add the chopped apples, brown sugar, and cinnamon, then cook for about 3–5 minutes, stirring occasionally, until the apples are slightly softened; remove from the heat and cool briefly.

Beat the softened cream cheese, granulated sugar, and vanilla extract until smooth and creamy.

Unroll both cans of crescent dough and separate the dough into triangles. Arrange the triangles in a ring on the prepared baking sheet, overlapping the wide ends in the center and pointing the narrow ends outward, leaving an opening of about 4 inches.

Spread the cream-cheese mixture around the wide center portion of the dough. Spoon the cooled apple mixture over the cream cheese, then drizzle or spoon about 1/4 cup of caramel sauce over the filling.

Fold each pointed dough end over the filling and tuck it underneath the inside edge of the ring. Press overlapping seams together if needed, and brush the dough with beaten egg if using.

Bake for 20–25 minutes, or until the crescent dough is puffed and golden brown.

Cool the ring for at least 10 minutes. Drizzle with the remaining caramel sauce before slicing and serving.