Boil the kalabasa pieces in a pot with water. Remove from heat when kalabasa is tender. Transfer to a bowl and purée, making about 1 cup of mashed kalabasa. Set aside.
In a large bowl, whisk the brown sugar and eggs until pale and foamy. Add the mashed kalabasa, olive oil, cinnamon, allspice, pepper, and kosher salt. Whisk until smooth.
Sift in the all-purpose flour, baking powder, and baking soda and whisk until well-combined.
Prepare a loaf pan and coat with butter or non-stick spray. Line it with parchment paper.
Pour the batter into the prepared pan and smooth the top of the batter with an offset spatula. Bake in a preheated 180°C oven for about 25-30 minutes or until the loaf has puffed. When done, set the loaf on a cooling rack to cool for ten minutes.
While kalabasa loaf is cooling down, combine the powdered sugar, olive oil, and a tablespoon of hot water in a bowl. whisk until smooth. Add more water if the glaze is too thick. Make sure the loaf is cooled before pouring the glaze on top. Serve when the glaze has set.