Mix bourbon and vanilla extract in a medium bowl and add pear slices, tossing gently to coat. Set aside to soak for about 20 minutes. Combine brown sugar, butter, and flour into small lumps; set aside for topping the galette.
Drain the pears (reserve the bourbon mixture for a cocktail*) and toss with the granulated sugar. Arrange pear slices in overlapping rows on the dough and sprinkle topping over the pears.
Pleat the dough, brush with egg mixture, and sprinkle with sugar. Bake for 30 minutes and then cover with tented foil and bake for 30 minutes longer.