Slice the peaches into ¼-inch wedges, discarding the pits. Place the sliced peaches in a medium bowl.
In a small bowl, whisk together the brown sugar, cinnamon, cornstarch, allspice, and nutmeg.
Sprinkle the spice mixture over the peaches and gently toss to coat. Add the vanilla extract, lemon juice, honey, and almond extract (if using). Fold gently until evenly combined, being careful not to break up the peach slices.
Line a large baking sheet with parchment paper.
On a lightly floured surface, roll the pie crust into approximately a 12-inch circle. Transfer it to the prepared baking sheet.
Arrange the peaches in the center of the crust, leaving about a 2-inch border around the edges. Discard any excess liquid remaining in the bowl.
Fold the edges of the crust up and over the peaches, overlapping sections as needed and gently pressing them together to seal.
Spoon about ½ tablespoon of the reserved peach juices over the fruit.
Refrigerate the assembled galette for 30 minutes.
Meanwhile, preheat the oven to 400°F (200°C).
Brush the crust with the beaten egg and sprinkle evenly with the cane sugar.
Bake for 36–38 minutes, rotating the baking sheet halfway through baking, until the crust is golden brown and the peach filling is bubbling.
Allow the galette to cool on the baking sheet for at least 15 minutes before slicing and serving.