Preheat the oven to 350°F and coat a 10-cup bundt pan with nonstick spray.
Core tomatoes and quarter. Purée in a blender or food processor until completely smooth. Measure out two cups and set aside, reserving remainder.
In a large bowl, beat together brown sugar, granulated sugar, and butter until creamed and fluffy.
Beat in eggs, one at a time.
In another bowl, sift together flour, baking powder, baking soda, salt, cinnamon, cardamom, and ginger.
Alternate adding flour mixture and tomato purée to the batter, starting and ending with the flour. Be sure to incorporate fully before doing the next addition.
Pour batter into prepared pan and bake for 40 minutes, or until a toothpick comes out clean.
Let cake cool in pan 10 minutes, then invert onto a rack and cool completely.
To make the glaze, mix together powdered sugar, vanilla, and 2-3 tablespoons leftover tomato juice until completely smooth. Add more sugar or liquid as necessary. Drizzle over cooled cake.