Preheat oven to 350 degrees F.
Melt butter in a small saute pan over medium heat until bubbling. Add the sprigs of thyme and continue cooking over medium heat until the thyme releases it scent, about 1 minute.
To the saute pan, add brown sugar and cook until foamy, stirring often to prevent the sugar from burning, and comes together and looks like caramel (but will grainy to the touch), about 2 minutes.
Working quickly, dump the hot caramel into a 10-inch cake pan and spread evenly. Allow to cool for 10 to 15 minutes.
Snuggly place the plums cut-side down over the caramel. Set aside.
Sift together flour, baking powder, cornmeal, and salt. Top with the fennel seeds.
Place the butter in a bowl of a stand mixer. With the paddle attachment, mix at medium speed for 1 minute. Add the granulated sugar, thyme, and lemon zest and mix for 5 minutes or until light and fluffy.
Add egg yolks, one at a time, mixing well after adding each yolk. Add the vanilla paste and anise liqueur.
Take the mixer down to low speed and add the flour alternating with the milk three times. Scrape the batter into a large mixing bowl and set aside.
Wash and dry the mixer thoroughly. Place the egg whites in the bowl and with a whisk attachment, mix on low speed for 30 seconds. Increase the speed and slowly sprinkle in the granulated sugar. Continue whipping until the egg whites form peaks, 2 to 3 minutes. The whites should look glossy, not dry or over whipped. Fold one-third of the whites into the cake mix. Once incorporated fold the remaining whites taking care not to overwork the whites.
Pour the cake over the plums and place in the oven. Set a timer for 30 minutes. Rotate the cake and bake for an additional 30 minutes more. To test for donenees, insert a toothpick into the center of the cake. If it doesn't come out clean, bake for an additional 3 to 5 minutes. Remove the cake from the oven and allow to cool on a cooling rack for 30 minutes.
Run a paring knife around the edges and invert onto a serving platter and garnish with fresh thyme sprigs.