Lightly grease the bottom and sides of a 9-inch springform pan with butter. Line the bottom and slightly up the sides with parchment paper and lightly butter the parchment. Set aside.

Preheat the oven to 325 degrees F.

In a large, heavy skillet, melt the butter over medium-high heat. Add 3/4 cup of the sugar, the water, and 2 tablespoons of the liqueur and bring to a boil. Lower the heat and cook until the sugar melts and starts to caramelize. Remove the pan from the heat and add the pears, stirring to coat with the caramel. Return the pan to medium heat and cook, stirring occasionally, until the pears are tender, 8 to 10 minutes, depending upon the ripeness. Remove from the heat and remove the pears with a slotted spoon to a baking dish. Return the pan to medium heat and cook until reduced to a thick syrup. Remove from the heat and let cool. Arrange the pears across the bottom of the pan. Pour the syrup over the pears. Let cool while preparing the batter.

In a large bowl, beat the cream cheese with the paddle attachment of an electric mixer, or with a large, heavy wooden spoon until creamy and smooth. Add the remaining 1 cup sugar a few tablespoons at a time, and when well incorporated, add the flour, the lemon peel, and remaining 4 teaspoons of liqueur. Add the eggs and sour cream, beating on low speed just until combined, scraping down the sides of the bowl to incorporate all the ingredients.

Pour the batter over the pears in the prepared pan and bake in the middle of the oven until set, but the center is still slightly soft (but not wobbly), about 1 hour to 1 hour and 10 minutes. Remove from the oven, loosen the outer ring, and let cool in the pan on a wire rack. Refrigerate the cake, loosely covered, for 6 hours or overnight.

Remove the cake from the refrigerator and remove the outer ring. Place a cake platter on top of the cake and quickly invert so that the fruit is on top. Carefully peel away the parchment paper. To serve, cut the cheesecake with a thin, sharp knife dipped in hot water. Arrange the slices on dessert plates with 3 shortbread cookies each and serve.