Preheat an oven to 375 degrees F. Butter or grease a jelly-roll pan. Press a 16-by-12-inch sheet of parchment or waxed paper into the pan with your fingers. Remove the paper, turn it over, and place it back into the pan, pressing it into place. (This will evenly grease both sides.) With scissors, trim off any excess paper. Butter, grease, or line two 2-inch center cups of a muffin tin. Set aside. To make the cake, in a large bowl, using the electric mixer set on high speed, beat the eggs until thick and lemony, about 5 minutes. Add the sugar and continue to beat on high for 5 minutes. On low speed, beat the water and vanilla. Sift together the cake flour, baking powder, and salt into the batter, and beat until just combined. Spoon 2 to 3 tablespoons batter into each cupcake cup. Pour the remaining batter into the prepared baking pan. Tilt the pan to evenly spread the batter. Place the baking pan in the oven and bake until the cake springs back when lightly touched and just begins to turn golden, about 10 minutes. Bake the cupcakes until they just begin to turn golden, about 10 to 12 minutes. Let the cake and cupcakes cool on a wire rack for 5 minutes. Set aside cupcakes. Lay a clean kitchen towel on a countertop. Sift confectioners' sugar over the towel. Invert the cake onto the towel and peel off the paper. Beginning at a long side, roll up the cake with the towel. Refrigerate the cake roll for 30 minutes. To make the filling, in a bowl, using the electric mixer set on medium-high speed, beat the cream until it begins to thicken. Add the vanilla and confectioners' sugar and continue to beat until firm peaks form and the mixture is spreadable. To make the frosting, in a large bowl, using electric mixer set on low speed, beat together the butter, cocoa, and salt until well mixed, about three minutes. Drizzle in the half-and-half and vanilla and continue to beat on low speed until blended, scraping the bowl with a spatula. Slowly add the confectioners' sugar, beating until blended. Increase the speed to medium and beat until light and fluffy, about 2 minutes. Scrape down the bowl and continue to beat for 1 minute. Cover with plastic wrap and let stand at room temperature. To assemble, remove the cake from the refrigerator and unroll it on a counter. Using a butter spreader or spatula, spread the filling on the top to within 1 inch of the edges. Beginning at a long side, carefully roll up the cake, removing the towel. Center 2 dollops of frosting along a serving platter where the cake will rest. Place the cake on top of the dollops, pressing down gently to secure it on the platter. Reserve 1/2 to 3/4 cup frosting for the cupcakes. Frost the cake roll with the remaining frosting. Frost the cupcakes and arrange on the cake roll at different angles to resemble cut branches. (You may want to cut one cupcake in half widthwise to give the branches a more natural look.) If desired, decorate with chocolate leaves (recipe follows) and small plastic forest animals and finish with evergreen sprigs.