Preheat your oven to 400°F (200°C).
Roll out the pie crust on a lightly floured surface to a 12-inch circle.
Layer the potato slices evenly on the crust, leaving a 2-inch border.
Sprinkle cheddar and Parmesan cheese over the potatoes, adding chives, salt, and pepper.
Fold the edges of the crust over the filling, pleating as you go to create a border.
Brush the exposed crust with the beaten egg.
Bake the galette in the preheated oven for 35-40 minutes or until the crust is golden brown and the potatoes are tender.
Let the galette cool for 5-10 minutes before slicing and serving.