Make the Topping:
Position a rack in the center of the oven and heat the oven to 350 degrees F. Butter the bottom and sides of a 9x2-inch round cake pan
(don't use a springform pan, as the caramel might leak out during baking). Line the bottom of the pan with a round of parchment and butter the top of the paper.
Peel, core, and cut the pears lengthwise into 1/4-inch-thick slices. Arrange the pear slices on the bottom of the pan in a circle around the edge, overlapping them slightly, with the pointed ends towards the center. If necessary, cut a little off the pointed ends to make the slices fit better. Or if the pear slices don't reach all the way to the middle, arrange a few of the shorter slices in the center to cover the bottom of the pan.
Make the Cake Batter:
Make the Basic Caramel according to the directions. Immediately remove the pan from the heat and whisk in the 4 tablespoons of butter one piece at a time, until they are completely melted. Carefully pour the hot caramel evenly over the pears (it should spread over the pears and onto the bottom of the pan).
Sift the flour
, baking powder, ginger, cinnamon, and salt into a medium bowl
. Stir to combine. In a small bowl, stir together the milk and vanilla.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium-high speed until light and fluffy, about 1 minute. Turn the mixer to medium and slowly add the brown sugar. Increase the speed to high and continue to mix until lightened in texture and color, 2 to 3 minutes total. Reduce the speed to medium and add the eggs one at a time, beating well after each addition and scraping down the sides of the bowl as needed.
Bake the cake:
Reduce the speed to low and alternate adding the flour mixture and milk mixture in five additions, beginning and ending with the flour. Mix each addition just enough to incorporate, as overmixing will lead to a tougher cake. Scrape down the sides of the bowl one last time and mix briefly to blend well.
Spoon the batter in large dollops over the pears and smooth it into an even layer with an offset spatula. Bake the cake until the top is golden brown and a toothpick inserted in the center comes out clean, 35 to 45 minutes. Transfer the pan to a wire rack to cool for 10 minutes. Run a knife around the edge of the pan. Turn a cake plate upside down on top of the cake pan and, using pot holders
, carefully invert the cake pan onto the plate.
Fill a cup
measure halfway with water and put a pastry brush in it; this will be used for washing down the sides of the pan to prevent crystallization. In a heavy-duty 2-quart saucepan, stir the sugar, lemon juice, and 1/4 cup cold water. Brush down the sides of the pan with water to wash away any sugar crystals.
Bring to a boil over medium-high heat and cook, occasionally brushing down the sides of the pan, until the mixture starts to color around the edges, 5 to 8 minutes.
Gently swirl the pan once to even out the color and prevent the sugar from burning in isolated spots. Continue to cook until the sugar turns medium amber, about 30 seconds more. (Once the mixture begins to color, it will darken very quickly, so keep an eye on it.)