In a large bowl, combine the flour, baking soda, baking powder, and salt.
Cut the butter into the dry ingredients until the mixture has the consistency of coarse sand.
In another bowl, combine the egg, honey, yogurt, starter, and vanilla.
Add the wet ingredients and the crystallized ginger to the dry ingredients, mixing just until everything is incorporated.
Preheat the oven to 350F.
Butter a 7.25-inch springform pan and line the bottom with a 9-inch circle of parchment (the parchment will come up the sides a bit.
In a small saucepan, combine the topping ingredients (except plums) and cook over medium heat until the butter is melted, then for another 2 minutes.
Pour the syrup into the parchment-lined pan. Arrange the plum slices in the pan over the syrup.
Spoon the dough and spread it evenly over the plums. It is easier to ensure that the dough reaches to the sides of the pan if you spoon it around the perimeter and spread inward rather than the other way around.
Bake for 30 minutes or until a toothpick emerges clean.
Cool in the pan on a wire rack for 10 minutes. Then loosen the springform, invert the cake onto a plate, and carefully peel away the parchment.