In a skillet, bring 1 cup of water and the brandy to a boil, then reduce heat to a simmer. Peel the pears with a vegetable peeler, cut in half lengthwise and remove the core, the stem and the fibers that run down the center. Slice lengthwise into 1/4-inch thick slices and add to the simmering brandy mixture. (Add enough hot water to cover the pears, if necessary.) Poach for about 10 minutes, remove from heat and let cool to room temperature while you are preparing the batter.
For the cake:
Preheat oven to 350 degrees F. Oil the bottom of a 9 by 2-inch deep round pan and lay parchment paper over the bottom. Then brush a thin layer of canola oil on top of the parchment paper. Sift cake flour and cocoa powder together several times to aerate. In a separate heatproof bowl combine the eggs and sugar. Nest the bowl with the sugar and eggs in another larger bowl filled with about 2-inches hot water. Whisk together the eggs and sugar, being very careful not to splash any water into it. The idea here is to raise the temperature of the egg and sugar mixture to between 100 and 120 degrees F. Use a candy thermometer to test the temperature. Once the mixture is warmed, remove the bowl from the water bowl and use an electric beater to beat it until it is about triple in volume and is the consistency of soft whipped cream. Sift the 1/3 of the flour and cocoa powder into the bowl of beaten eggs and sugar, and fold together. Repeat for each remaining 1/3 of the flour mixture, folding together after each addition. Then, gently fold in the melted butter and vanilla.
Drain the pear slices and arrange them in the bottom of the cake pan. Overlap them in a pattern of concentric circles. Pour batter over layer of pears, spreading evenly, but don't fill pan any more than within 1/2-inch of the top edge of the pan. Bang pan on countertop - 1 or 2 times only - to make sure the batter sinks down between the pears, but don't overdo it or you will undo all your hard work in aerating the batter! Bake in oven until the surface springs back when touched and a toothpick inserted in the cake comes out clean, about 20 to 25 minutes. Let cake cool in pan 10 minute to allow set up, then using oven mitts, press a serving plate firmly on the top edges of the pan, and invert the cake onto the plate. Replace any pear slices that become dislodged. This cake can be served warm or at room temperature.