Preheat your oven to 400°F (200°C). Slice the pears thinly, aiming for even pieces to ensure they cook uniformly. Chop the pecans roughly and set aside.
In a small saucepan, melt the 1/4 cup unsalted butter over medium heat. Once melted, stir in the brown sugar, vanilla extract, cinnamon, and salt. Allow the mixture to bubble gently for 2-3 minutes, stirring occasionally, until the sugar dissolves and the sauce thickens slightly. Remove from heat and stir in the chopped pecans. Set aside to cool slightly.
Unroll the refrigerated pie crust onto a parchment-lined baking sheet. If desired, gently roll out the crust to a roughly 12-inch circle. Arrange the sliced pears in the center, leaving about a 2-inch border around the edges. Pour the caramel pecan mixture evenly over the pears.
Fold the edges of the dough up and over the pear filling, pleating gently to create a rustic border. Make sure some filling peeks through the center.
Brush the folded crust edges with the beaten egg. This step will give your Caramel Pecan Pear Galette a beautiful golden and glossy finish when baked.
Place the galette in the preheated oven and bake for 30-35 minutes, or until the crust is golden brown and the pears are tender and bubbling. Keep an eye on the crust to avoid over-browning; if it browns too quickly, tent it loosely with foil.
Allow the galette to cool for at least 15 minutes before slicing. This resting time lets the caramel set slightly, making it easier to serve. Enjoy warm, optionally with a scoop of vanilla ice cream or a dollop of whipped cream.