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Preheat oven to 350F/180C. Butter and flour a 9x5-inch loaf pan or an 8-inch square or round pan, or line the bottom and sides with parchment paper (for an easy release as the cake is sticky).
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In a medium saucepan over medium heat, bring the stout to a boil. Remove from heat and carefully add baking soda (the mixture will foam). When the foaming subsides, add molasses, granulated sugar, and brown sugar and mix to combine. Set aside.
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In a large bowl, sift together flour, baking powder, salt, ground ginger, cinnamon, cloves, nutmeg, and cardamom. Set aside.
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Add eggs, oil, and grated ginger to the stout mixture and whisk until combined. Add stout mixture to flour mixture in 3 additions, whisking between each addition, until smooth.
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Scrape batter into prepared pan and smooth the top. Bake until the top springs back when gently pressed and a toothpick inserted into the center of the cake comes out clean – 35-45 minutes for an 8-inch pan, or 45-55 minutes for a loaf pan. Do not open the oven until the gingerbread is almost done or the center may fall slightly. Transfer pan to a wire rack to cool completely. Serve at room temperature.
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Store cake at room temperature or in the refrigerator in an airtight container for up to 4 days. Cake can also be frozen for up to 2 months. Thaw on the counter, still covered, or overnight in the fridge.