01 - Preheat oven to 350°F. Line a 10x15-inch jelly roll pan with parchment paper and lightly grease the surface.
02 - In a mixing bowl, beat the eggs with granulated sugar and vanilla extract using an electric or stand mixer for 5 minutes, until the mixture turns pale and triples in volume.
03 - Sift together the all-purpose flour, cocoa powder, baking powder, and salt. Gently fold the dry mixture into the beaten eggs until just combined, being careful not to deflate the batter.
04 - Pour the batter into the prepared pan and spread it into an even layer. Bake for 10 to 12 minutes, or until the cake springs back lightly when pressed with a fingertip.
05 - While still hot, invert the cake onto a clean kitchen towel dusted with powdered sugar. Peel away the parchment paper. Starting from the short end, roll the cake and towel together into a tight log. Allow to cool completely in this shape.
06 - Once cooled, carefully unroll the cake. Spread the caramel sauce evenly across the surface, reserving about 2 tablespoons for finishing. Sprinkle the chopped toasted pecans in an even layer over the caramel.
07 - Gently re-roll the cake without the towel, keeping it snug but not too tight. Place the roll seam-side down on a serving platter.
08 - Place the chopped semi-sweet chocolate in a heatproof bowl. Heat the heavy cream until just simmering, then pour it over the chocolate. Let sit for 2 minutes, then stir until smooth and glossy. Allow to thicken slightly.
09 - Pour the ganache over the cake roll, letting it cascade down the sides. Drizzle with the reserved caramel sauce and arrange pecan halves on top.
10 - Refrigerate for 20 minutes to set the ganache before slicing. Allow the cake to come to room temperature before serving for the best texture.