For the cake: Preheat the oven to 350 degrees F. Line the bottom and sides of a 9-by-5-inch loaf pan with parchment paper.
Sift the flour, baking powder, baking soda, cinnamon, salt, cloves and ginger into a medium bowl.
In a large bowl, use an electric mixer to cream together the granulated sugar, brown sugar, butter and vanilla. Beat in the eggs. Alternate adding the flour mixture with the cider, about half of each at a time, and mix just until combined.
In a small bowl, toss the apples with the lemon juice. Stir into the batter. Scoop the batter into the prepared pan. Bake until a toothpick inserted into the center comes out clean, 50 to 60 minutes.
Let cool for at least 20 minutes before lifting the parchment paper to remove the cake from the pan.
For the sauce: Cook the apple cider and lemon juice in a large heavy-bottomed saute pan set over medium heat until reduced to a syrupy consistency, about 10 minutes. Stir in the heavy cream and bring back to a boil. Reduce the heat to low and whisk in the butter, 2 pats at a time, allowing the butter to melt before adding more. Serve warm with the apple cider cake.
Recipe courtesy of Kimberly Schlapman