Preheat the oven to 350 degrees Fahrenheit with a rack in the center position. Grease a sheet pan with nonstick spray, line it with Gefen Parchment Paper, and grease the parchment, too.

In a medium bowl, whisk the flour, baking powder, baking soda, cardamom, nutmeg, and salt.

In the bowl of a stand mixer fitted with the whisk attachment or in a large bowl with a handheld mixer, whip the eggs and granulated sugar together on high speed until light yellow and doubled in volume, about five minutes.

In a small bowl or large, spouted measuring cup, stir together the maple syrup, oil, and vanilla.

With the mixer running on medium speed, slowly pour the syrup mixture into the egg mixture and whip until incorporated, about two minutes. Add the dry ingredients and mix on low speed until just combined, then fold in the shredded squash with a rubber spatula.

Pour the batter into the prepared pan and spread it evenly to the corners with a large offset spatula. Bake for 25 to 30 minutes, until the cake is deeply golden brown and a tester inserted into the center comes out clean.

While the cake bakes, line a work surface with a large sheet of parchment paper.

Set the sheet pan on a wire rack and let the cake cool in the pan for about 10 minutes, then run a paring knife around the edges of the pan to loosen the cake. Invert the cake onto the parchment- lined work surface, carefully remove the baked-on parchment from the cake, and allow to cool completely, about 15 minutes.

In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl with a handheld mixer, beat together the maple syrup, butter, brown sugar, confectioners’ sugar, salt, and milk on medium-high speed until smooth and spreadable, three to five minutes.

To make a rectangular two-layer cake, use a sharp knife to slice the sheet cake in half crosswise. Set one half of the cake on a serving platter and spread with a generous layer of frosting, then place the second half of the cake carefully on top. Spread the rest of the frosting over the top of the cake, leaving the sides bare.

Slice and serve the cake immediately. The cake will keep, uncovered, in the refrigerator for up to three days.