Preheat oven to 325ºF. Put the fruit in a small saucepan and cover with water (or brandy or rum). Heat to a simmer; turn off the heat and let the cherries soak for about 20 minutes. Drain (saving the liquid if it is alcohol).
Prepare the baking pan. You can make this upside-down cake in an 8-inch cast-iron skillet (see Kitchen Notes if you take this route), or in an 8-inch round cake pan. I couldn’t find either of those things anywhere (I told you I was really cold), but I did find the vaguely 8-inch square baking dish I use for gingerbread. It worked very well for this. Prepare the pan by coating lightly with oil.
Prepare the fruit. Put the butter and brown sugar in a small skillet—heat to melt the butter, then stir and simmer for three or four minutes. Then stir in the drained cherries and simmer everything together for a minute or two. If you are using a cast-iron skillet, you may start the fruit in it.
Grind the star anise into a powder—you can use a small electric spice or coffee mill, or do it by hand with a mortar and pestle. Our coffee grinder is getting ancient and did a pathetic job of the grinding, so I ended up finishing the process by hand, with the mortar. You want 2 teaspoons of star anise.
Put the star anise in a medium-sized mixing bowl. Sift in the flour, add the white sugar, baking powder and salt.
In another mixing bowl, pour in the olive oil and add the yogurt and almond extract. Whisk together. Then add the eggs and whisk until everything is smooth and uniform.
If you are using a baking pan rather than a cast-iron skillet, pour the fruit mixture into the bottom of the pan and distribute it evenly.
Add the wet into the dry ingredients a little at a time, stirring with a wooden spoon or silicone spatula until everything is just mixed together. The batter will be rather thick. Pour it over the fruit and spread evenly.
Bake about 45 minutes or until the top is lightly golden and a thin blade inserted near the center comes out nearly dry. Remove from oven and cool on a rack for 10 minutes. Run a thin blade around the edge. To remove the cake from the pan, put a plate face down on top of the pan and flip over the whole shebang onto a serving plate. Serve warm or let come to room temperature.