1. Preheat oven to 300 degrees F. Grease a 9-by-5-by-3-inch loaf pan with nonstick cooking spray.
2. Cook rice according to the package directions.
3. In a nonstick skillet, melt butter over medium heat. Turn off the heat and stir in brown sugar until sugar is dissolved. Transfer the mixture into the prepared loaf pan tilting the pan so that the mixture is distributed evenly.
4. Cut apple into four, core and slice each quartered apple thinly lengthwise. Place apple slices overwrapping over the brown sugar butter mixture.
5. In a medium bowl, whisk eggs then whisk in yogurt, milk, granulated sugar, lemon juice and vanilla. Whisk in rice until combined. Put the mixture over the apple slices. Put the loaf pan into the oven and bake for 60-65 minutes or until the edges are lightly browned and a toothpick inserted in center of the cake comes out with a few moist crumbs attached. Turn off the heat and let the pan sit in the oven for 20 minutes.
6. Remove the pan from the oven, let cool for about 15 minutes then refrigerate until completely chilled (at least 6 hours, ideally overnight).
7. To serve, invert the pan onto a large plate or cutting board. Sprinkle with cinnamon if you like. Slice the cake into your desired size.