Stir peaches, 1/2 cup sugar, and salt together in a bowl. Let sit for 20 to 30 minutes. Strain accumulated juices into a saucepan and return strained peaches to bowl.
Bring juices in saucepan to a boil; reduce heat to medium-high, and cook until mixture thickens to a syrup, 5 to 10 minutes.
Whisk 1/2 cup sugar, flour, and cornstarch together in a bowl. Stir lemon juice, cayenne pepper, and cinnamon into peaches; add flour mixture and toss to coat. Pour peach syrup over peach mixture and stir.
Roll out half the pastry to fit a 10-inch pie plate. Place bottom crust in pie plate. Pour peach filling into pie shell, tap pie plate against a surface to settle filling, and dot with butter.
Roll remaining crust into an 11-inch circle and cut into 1-inch strips using a paring knife or pastry wheel.
Lay 5 strips of pie crust over the top of the pie. Weave remaining pastry dough strips through to form a lattice top. Fold the ends of the lattice strips under the bottom crust and crimp the crust. Brush cream over the top and sprinkle with 1 tablespoon sugar.
Bake in the preheated oven until browned and bubbling, about 1 hour 15 minutes. Cool completely before slicing.