For the French Apple Cake:
Place sliced apples in a microwave-safe shallow bowl. Microwave in 1-minute intervals for about 3 minutes, stirring in between, until softened and pliable. Toss with lemon juice and spiced rum, then set aside to cool for about 10 minutes.
Preheat oven to 325°F. Butter a 9-inch springform pan and place it on a parchment or silicone mat-lined baking sheet (important as the batter may leak).
In a medium bowl, whisk together 1 cup flour, 1 cup sugar, baking powder, and salt.
In a large bowl, mix the whole egg, milk, oil, and vanilla bean paste. Gradually add the dry ingredients and mix until just combined. Reserve 1 cup of this batter.
To the remaining batter, whisk in 2 egg yolks and fold in the cooled apple slices. Pour the apple mixture into the prepared pan, pressing the apples down to create an even layer.
Add 2 tbsp of flour to the reserved batter and whisk until smooth. Spoon this batter over the apple mixture and spread evenly. Sprinkle the remaining 1 tbsp of sugar over the top.
Bake for 1 hour to 1 hour and 15 minutes, until golden brown. Let cool on a wire rack, then remove from the pan once completely cooled. Dust with powdered sugar before serving.
For the Maple Ginger Custard Sauce:
In a small saucepan over medium heat, combine milk, maple syrup, and sliced ginger. Bring to a simmer, then remove from heat and let steep for 30 minutes.
After steeping, strain out the ginger slices and return the milk mixture to a simmer.
In a small bowl, whisk the egg yolks. Slowly whisk in a bit of the warm milk mixture to temper the eggs, then add the yolk mixture back into the saucepan. Stir constantly until the custard thickens and coats the back of a spoon.
Remove from heat and stir in vanilla bean paste and spiced rum. Pour into a heat-safe bowl, cool on the counter, then refrigerate until cold.