Preheat the oven to 350°F (175°C). Thoroughly grease a 9-inch round cake pan.
Distribute the 1/3 cup brown sugar or maple sugar evenly across the bottom of the prepared pan. Pour the 3 tablespoons melted salted butter over the sugar.
Arrange the prepared pear slices over the butter and sugar layer, positioning them decoratively and allowing the slices to overlap slightly if desired.
In a separate bowl, thoroughly whisk together the whole wheat pastry flour, baking powder, salt, cinnamon, ginger, freshly grated nutmeg, and ground cloves. Set the dry mixture aside.
Place the 6 tablespoons softened salted butter and 3/4 cup maple sugar in a large mixing bowl. Beat them together until the mixture becomes pale, light, and fluffy.
Add the egg yolks individually, mixing thoroughly after each addition. Blend in the vanilla extract.
Add approximately half of the prepared flour mixture to the butter mixture and stir until incorporated. Mix in the whole milk, followed by the remaining dry ingredients. Stir only until the batter is evenly combined.
In a completely clean and dry bowl, beat the reserved egg whites together with the cream of tartar until firm, stiff peaks develop.
Carefully fold the beaten egg whites into the cake batter, working gently so the incorporated air is retained.
Transfer the completed batter to the cake pan, spreading it carefully over the arranged pears. Level the surface evenly.
Bake for 35–40 minutes, until the cake is golden and a toothpick inserted into the center comes out clean.
Remove the cake from the oven and leave it in the pan for 10 minutes. Run a knife carefully around the perimeter to loosen the edges, then invert the cake onto a serving plate so the pear layer is facing upward.
Serve warm or at room temperature, either plain or accompanied by whipped cream, vanilla ice cream, or vanilla custard.