Place walnuts, flour, salt, baking powder, and baking soda in bowl of food processor. Pulse 8-10 times, until walnuts are ground finely. Transfer this dry mixture to a bowl.
Place the eggs and sugar in the bowl of the processor and process for 2 minutes. The mixture will become a pale yellow. Once the 2 minutes have passed, and while the processor is still running, pour a steady stream of the melted butter and oil through the feed tube into the bowl, continuing to process until it is mixed well.
Add the dry flour-walnut mixture from the bowl to the processor. Pulse 4-5 times to combine the wet and dry ingredients.
Pour the walnut batter over the prepared pears in the baking pan, spreading the batter evenly by slightly tipping the pan and gently thumping the bottom. If you see some pear pieces peeking out, don’t worry. The batter will rise over them as the cake bakes.
Bake the cake for 1 hour and 10 minutes. If you remember, rotate the pan halfway through. (I forgot and it seemed ok.) Test with a toothpick in the center to see if it comes out clean. (It may need another 5-10 minutes, depending on your oven.)
Remove from oven, place on a wire rack and allow to cool for 15 minutes.
Run a knife around the edges of the pan to ensure that the cake will loosen. Turn the cake onto another wire rack set within a rimmed baking sheet (this will catch any dripping goodness). Carefully take off the parchment paper.
Allow the cake to cool for around 2 hours. Transfer to a platter, cut into serving pieces and enjoy with whipped cream, ice cream, or crème fraiche.