Peel 2 of the pears and grate them coarsely for the dough.
Beat the eggs and sugar with a mixer very well until whitened.
Mix the baking powder with the yoghurt, stir until effervescent, then add it to the eggs.
Pour the oil in a trickle, while beating on a low setting. Next add the sifted flour.
Add the grated pears and vanillas to the mixture, while stirring with a ladle.
Pour the mixture into a nonstick form about 11 3/4″ (30 cm) in diameter or one laid out with baking paper.
Slice the other 2 pears thinly and arrange them randomly on top of the mixture, without pressing down on them. Sprinkle 2 tbsp brown sugar over the pears.
Bake in a preheated 356°F (180 °C) oven about 50 min.
Sprinkle the cooled cake lightly with powdered sugar.