Preheat the oven to 350°F (175°C). Grease a 9-inch cake pan and line the bottom with parchment paper.
Heat a large skillet over medium heat. Add 1 tablespoon of butter. Once melted, add the sliced apples and a pinch of salt. Sauté for 5-6 minutes, until the apples begin to caramelize.
Add the maple syrup to the skillet and bring to a simmer. Reduce the heat and simmer for 2 minutes, then turn off the heat. Transfer the apples and syrup to a bowl to cool slightly. Wipe out the skillet.
Add the remaining stick of butter to the skillet over medium heat. Let it melt and then turn the heat to low, allowing the butter to brown slightly and become fragrant. Remove from the heat and let cool.
In a mixing bowl, whisk together the flour, sugar, baking powder, and salt until combined. In a separate bowl, beat the eggs and whisk in the ricotta, browned butter, and vanilla extract.
Gradually fold the dry ingredients into the wet mixture in two batches, using a rubber spatula to avoid over-mixing. Fold in ¾ of the apples and all of the maple syrup from the bowl.
Pour the batter into the prepared cake pan and smooth the top. Arrange the remaining apple slices on top of the batter.
Bake the cake on the middle rack for 40-45 minutes, or until a toothpick inserted into the center comes out clean. Remove from the oven and let the cake cool for 10 minutes.
After 10 minutes, lift the cake out using the parchment paper. Slice and serve warm or at room temperature with a sprinkle of raw sugar and a drizzle of warm maple syrup.