Preheat oven to 350 degrees. Generously grease a tart pan with butter. Coat in flour. Make sure to shake off all excess flour.
Add 1 1/2 sticks (170g) of butter to the bottom of a stand mixer. Beat on medium speed until light and fluffy, about 2 minutes. Add sugar, beat another 2-3 minutes. Add vanilla and beat just until combined. Add flour and salt, and mix on low speed, just until combined.
Transfer dough to prepared pan, and use fingers and palm of hand to press the dough into the bottom of the pan and up the sides. Bake until lightly golden brown and puffed, about 15-20 minutes. (Start checking at 15 minutes.) Let the pie crust cool completely.
While the crust cools, make the filling and pear topping. Add mascarpone, powdered sugar, 1 tsp. vanilla bean paste, and 1/2 teaspoon almond extract to the bottom of a stand mixer. Beat on medium speed until combined, set aside.
Heat a large non-stick pan to a medium-high heat. Add remaining 2 tablespoons unsalted butter. Once the butter melts, let it begin to brown. Once it turns brown, start to swirl the pan. Continue to swirl the pan until the butter turns a deep, brown color and smells nutty. Turn the heat down and add pears.
Coat pears in the brown butter, and cook until pears are slightly softened, about 3-4 minutes. Add brown sugar. Continue to cook until pears are soft, another 3-4 minutes. Remove from heat and add vanilla bean paste. Toss until coated. Transfer to a plate or bowl and cool completely.
To assemble: Spread vanilla mascarpone mixture into the bottom of the cooled shortbread crust. Arrange the pears slices artfully on top. Spoon any accumulated juices on top of the pears. Sprinkle with flake sea salt if desired.
Serve at room temperature or chilled.