Position a rack in the lower third of the oven and preheat to 450 degrees F. Butter a baking sheet with 1 tablespoon butter. Unfold 1 sheet puff pastry on a floured surface. Beat the egg with 1 tablespoon water and brush over the dough. Unfold the other sheet of puff pastry and lay it on top; press the sheets together with a rolling pin, then roll out into a 14-inch square. Transfer to the prepared baking sheet and refrigerate until firm, about 30 minutes. (Refrigerate the leftover egg wash for brushing the galette.)

Combine the sugar, allspice, vanilla, lemon zest and lemon juice in a large bowl; transfer 2 tablespoons of the mixture to a small bowl and set aside for sprinkling. Add the cherries, peaches and 3 tablespoons flour to the remaining sugar mixture and toss.

Trim the edges of the pastry with a pizza wheel or sharp knife to make a 12-inch round. Spoon the cherry mixture in the center, leaving a 2-inch border. Cut the remaining 1 tablespoon butter into small pieces and scatter on top. Fold in the pastry edges over the filling, pleating as needed.

Bake the galette until puffed and golden, about 20 minutes. Reduce the oven temperature to 350 degrees F. Remove the galette from the oven, brush the crust with the reserved egg wash and sprinkle with the reserved sugar mixture. Return the galette to the oven and bake until the juices thicken and the pastry is cooked through, 30 to 45 more minutes (depending on the juiciness of the cherries). Transfer to a rack and let cool at least 10 minutes so the filling sets.

Photograph by Andrew Purcell