Preheat your oven to 400°F.

Unroll the refrigerated pie crust onto a sheet of parchment paper. Lightly dust a rolling pin with flour and roll the crust into a 12-inch circle.

Carefully lift the parchment paper with the crust and transfer it to a rimmed half-sheet pan.

In a medium mixing bowl, combine the lemon juice and vanilla extract. Whisk in the cornstarch until smooth, then whisk in the granulated sugar, brown sugar, and cinnamon.

Place the sliced peaches in a large bowl and pour the lemon-sugar mixture over them. Use a spatula to gently toss until the peaches are evenly coated.

Arrange the coated peach slices on the crust in a circular pattern, leaving a 2-inch border around the edge. Pour any remaining juices from the bowl over the peaches.

Fold the outer edge of the crust over the peaches, pleating and crimping as needed to create a rustic border.

Dot the peaches with the small pieces of butter.

In a small bowl, whisk together the egg and milk to make an egg wash. Brush the egg wash over the exposed crust, then sprinkle the coarse sugar evenly over the crust.

Place the sheet pan in the preheated oven and bake for 25 to 30 minutes, until the crust is golden brown and the filling is bubbly. Check at 25 minutes; if not golden, bake for up to 5 more minutes.

Remove from the oven and let the galette rest on the pan for 5 minutes before slicing and serving.