With the rack in the middle position, preheat the oven to 350°F (180°C). Line a baking sheet with a silicone mat or parchment paper. Butter the sides of two 8-inch (20 cm) springform pans and line the bottoms with parchment paper.
Place the squash cubes on the baking sheet. Cook for 45 minutes, stirring twice.
In a food processor, purée the squash until smooth. Let cool. You will need 1 ½ cups (375 ml) of the squash purée for the cake.
In a bowl, combine the flour, baking powder, baking soda and spices.
In another bowl, cream the butter and brown sugar with an electric mixer. Add the eggs, one at a time, then the squash purée. Whisk until smooth. Add the flour mixture. Divide the batter between the two cake pans.
Bake for 30 minutes or until a toothpick inserted in the centre of one of the cakes comes out clean. Let cool. Unmould and let cool completely on a wire rack, about 2 hours.
Meanwhile, place the chocolate in a bowl.
In a small pot, bring the cream to a boil. Pour over the chocolate. Let sit for 2 minutes without stirring.
Whisk until the ganache is smooth and creamy. Let cool to room temperature, stirring occasionally, until thick and spreadable, about 3 hours.
Cut each cake in half horizontally to get four layers in total. Divide one-third of the ganache between three of the cake layers. Stack the ganache-covered layers one on top of the other. Top with the remaining cake layer. Completely cover the cake with the remaining ganache.