Position a rack in the center of the oven and preheat to 350 degrees. Lightly grease a 13-by-18-inch jelly-roll pan; line with parchment paper, letting it come up the short sides by about 2 inches. Brush the paper with oil or melted butter and lightly dust with flour.
For the filling, whisk the evaporated milk, 1 cup sugar, 3 egg yolks and the cubed butter in a saucepan over medium heat. Cook, stirring constantly, until thick and creamy, about 10 minutes. Remove from the heat; stir in 1 teaspoon vanilla, the coconut and pecans. Set aside to cool completely, stirring occasionally.
Meanwhile, whisk the flour, baking soda and salt in a medium bowl. Whisk the cocoa and chocolate in another bowl. Add the hot coffee to the cocoa mixture, stirring until smooth; stir in the buttermilk and the remaining 2 teaspoons vanilla.
Using a mixer, beat the softened butter and the remaining 1 1/4 cups sugar in a large bowl on high speed until fluffy, about 4 minutes. Reduce the speed to low; add the 3 whole eggs, 1 at a time, incorporating each before adding the next. Add the flour mixture in 3 parts, alternating with the cocoa mixture in 2 parts, beginning and ending with the flour. Beat until just blended; if necessary, fold the batter together with a rubber spatula.
Spread the batter evenly in the prepared pan. Bake until the cake springs back when pressed lightly, about 16 minutes. Cool slightly in the pan.
Run a knife around the edge and slide the cake with its paper onto a work surface. Spread the coconut filling evenly over the cake. Roll the cake from one of the short ends, pulling it away from the parchment into a tight cylinder; transfer to a platter. Cool completely and dust with the confectioners' sugar or cocoa.