Preheat the oven to 350F. Butter a 9x9 square baking dish and line with a sheet of parchment paper with long ends. This will allow you to remove the cake from the pan later for easier slicing and serving.
Cream the butter and sugar together until light and fluffy.
Beat in the eggs, one at a time, allowing the first to get completely incorporated before adding the second. Scrape down the sides of the bowl as necessary to get everything well blended.
Blend in the oil, spice, and extract. Then the salt and baking powder.
Fold in the flour until smooth, but don't over beat. Spread into your prepared pan and smooth out evenly.
Arrange the plum halves, cut side up, in a grid pattern across the surface of the cake, pushing them down just slightly into the batter.
Sprinkle the surface with the granulated sugar. Bake for 40-43 minutes, or until the cake is puffed and golden and a toothpick inserted into the cake (not the plums!) near the center comes out without wet batter on it. If the cake seems to be browning too quickly toward the end of baking lay a loose piece of foil over the top. Note: this cake is moist, especially right around the plums, so use your judgement and don't over bake.
Let the cake cool for 15 -20 minutes in the pan before using the parchment paper sling to lift it out to a cooling rack to finish cooling. I like to loosen the edges first with a thin offset spatula to help it come out easily.
Slice the cooled cake into squares when ready to serve. Store leftover loosely covered on the counter.