Preheat the oven to 350 degrees F (175 degrees C).

Stir together peaches, 1/2 cup sugar, and salt in a large bowl until combined; let sit for 20 to 30 minutes. Strain accumulated juices into a saucepan and return strained peaches to the bowl.

Bring juices in the saucepan to a boil; reduce heat to medium-high and cook until mixture thickens to a syrup, 5 to 10 minutes.

Whisk together 1/2 cup sugar, flour, and cornstarch in a medium bowl. Stir lemon juice, cayenne pepper, and cinnamon into peaches; sprinkle with flour mixture and toss to coat. Pour hot peach syrup over peach mixture and stir.

Roll out one pastry into a 12-inch circle. Press into the bottom of a 10-inch pie plate. Pour peach filling into crust, tap the pie plate against a flat surface to settle filling, and dot with butter.

Roll remaining pastry into an 11-inch circle and cut into 1-inch strips using a paring knife or pastry wheel.

Lay 5 pastry dough strips evenly over the top of pie. Weave remaining pastry dough strips through to form a lattice top. Fold the ends of lattice strips under bottom crust and crimp crust to seal. Brush cream over the top and sprinkle with 1 tablespoon sugar.

Bake in the preheated oven until crust is browned and filling is bubbly, about 1 hour 15 minutes. Cool completely before slicing.