Whisk egg and 2 teaspoons vanilla extract together in a small bowl.
Combine butter, white sugar, and 3 tablespoons confectioners' sugar in a large bowl; beat with an electric mixer until fluffy. Beat in egg mixture. Add flour and salt gradually, beating until dough comes together.
Turn dough out on a large piece of plastic wrap; wrap and chill until firm, at least 2 hours.
Preheat oven to 350 degrees F (175 degrees C). Line a rimmed baking sheet with parchment paper; spread dough in an even layer on top.
Bake in the preheated oven until center is firm and edges are golden, 10 to 15 minutes. Let cool completely.
Whisk cream cheese, 1/4 confectioners' sugar, and 1 tablespoon vanilla extract in a bowl until smooth. Spread icing over the cooled crust.
Cover icing with sliced pears, overlapping slices slightly.
Combine 3 tablespoons brown sugar and red wine in a small saucepan over low heat; stir until brown sugar is dissolved, 3 to 5 minutes. Add cinnamon and ginger. Bring to a boil; reduce heat and simmer until reduced to a syrup, 3 to 5 minutes.
Coat pear slices with syrup using a pastry brush. Sprinkle remaining 2 tablespoons brown sugar on top.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Broil pizza until brown sugar melts and browns, about 3 minutes. Grate fresh nutmeg over pizza; cut into squares.