For the wafers: Preheat oven to 350 degrees and place rack in center of oven.
On 2 sheets of parchment paper, using a pencil and ruler, draw a 9-inch square.
In a saucepan, warm the butter and oil over medium heat until just melted, then set aside to cool a bit. In a small bowl, whisk together the flour, cocoa, espresso powder (if using) and salt.
Into the butter mixture, whisk the sugar, then the beaten egg, combining thoroughly. Using a spatula, stir in the flour mixture, then the vanilla, mixing until smooth.
Divide the batter between the parchment paper squares; there will be slightly more than 3/4 cup of batter for each. With a rolling pin, roll the batter to fill the penciled outline, taking care to reach the corners. It will be quite thin. (You can also pat out the batter with your fingers.) Use the ruler to square off the edges. (Batter will not spread while baking.)
Slide 1 parchment onto a baking sheet and bake for 8 minutes, or until just firm. When first wafer is done, slide the wafer, paper and all, onto a counter, then repeat with the second wafer. If necessary, trim the edges along the pencil line with a pizza cutter or sharp knife. (The slim shards are the cook's treat.) Repeat this with the second wafer, then let both cool completely, still on the parchment paper.