For the cake:
2 to 3 ripe pears
11 tablespoons unsalted butter, divided
1 1/4 cups brown sugar, divided
1 1/2 cups all-purpose flour
1 1/2 teaspoon baking powder
1/2 teaspoon baking soda
1 teaspoon cinnamon
1 teaspoon ginger
1/2 teaspoon cardamom
1/4 teaspoon salt
1/2 cup unsalted butter, softened
1/4 cup sugar
1 1/2 teaspoons vanilla extract
2 large eggs
1/2 cup buttermilk
For the cinnamon butterscotch:
4 tablespoon unsalted butter
3/4 cup brown sugar
1 1/2 teaspoon cinnamon
1/2 cup heavy cream, divided
1/2 teaspoon vanilla extract
1/4 teaspoon salt, or to taste
For the whipped cream:
2 cups cold heavy whipping cream
1/4 cup granulated sugar
1 teaspoon vanilla extract
3/4 teaspoon ground cardamom, or to taste
Preheat oven to 350°F. Sift together the dry ingredients for the cake and set aside. Peal and core the pears. Slice into thin wedges, about 1/4" thick.
Place 3 tablespoons butter and 3/4 cup brown sugar in a 12-inch ovenproof skillet over medium-high heat. Stir until sugar starts to dissolve. Simmer over low heat, undisturbed, until thick and caramel-y, about 3 minutes. Remove from heat. Carefully arrange the pear slices on top of the caramel in a fan. Add more pears to the center of the skillet to cover the entire surface if needed.
Cream the remaining 8 tablespoons of butter, 1/2 cup brown sugar, and 1/4 cup granulated sugar together with an electric mixer. Scrape down the sides and bottom of the bowl. With the mixer on low, add in the vanilla and eggs, one at a time. Alternating between dry and wet, mix in the flour mixture and buttermilk in two to three batches.
Pour the cake batter on top of the pears in the skillet. With a small off-set spatula, smooth the batter over the pears. Bake about 25 to 40 (depending how large your skillet is) or until a toothpick inserted into the cake comes out clean. Let cake cool on a wire rack for 15 minutes. Place a cake plate or cutting board on top of the skillet. Carefully, using oven mitts, flip the skillet over and invert the cake onto the plate.
To make the cinnamon butterscotch, place the butter, sugar, cinnamon, and 1/4 cup of cream in a saucepan over medium heat. Stir until butter melts and sugar dissolves. Let come to a gentle boil and cook for 5 minutes. Remove from heat. While whisking, pour in the remaining cream and vanilla. Add salt to taste.
Before serving, whip the cream with an electric mixer. Once the cream begins