Preheat oven to 350F. Grease the cake pan with butter or oil. Line with parchment paper. Grease the parchment paper, too.
Halve the plums and carefully pry out the pits. Arrange in the bottom of the cake pan cut-side down.
In a small saucepan, add ½ cup sugar with 3 tbsp water. Cook over medium-high heat, swirling occasionally, until the sugar turns into a deep amber caramel. Quickly pour evenly over the plums in the cake pan. Set aside.
In a medium bowl, beat the butter with 1 cup sugar (either by hand or with a hand mixer using medium-high speed) until lightly and fluffy. Beat in the eggs one at a time. Add amaretto, too, if using.
Sprinkle over baking powder, salt, cinnamon, and flour. Mix in until just incorporated. Spread batter evenly oven the plums. Bake for 30-35 minutes, or until cake feels firm to the touch, is deep golden, and a toothpick when inserted comes out almost completely clean.
Let cake cool for 15-20 minutes. Run a butter knife around the edge of the pan. Carefully flip the cake onto a serving platter. Reveal the beautiful cake, peel off parchment if necessary and serve with vanilla ice cream.
