Preheat the oven to 425 degrees F (220 degrees C).

Melt butter in a saucepan. Stir in flour to form a paste. Add white sugar, brown sugar, lemon juice, and cinnamon; bring to a boil. Reduce heat; simmer until sugar dissolves.

Line a 9-inch pie plate with 1 crust. Fill with sliced persimmons, mounding them slightly in center. Pour sugar-butter mixture over persimmons.

Sprinkle cinnamon sugar over remaining pie crust. Cut into strips. Make an X with the 2 longest strips in center of pie. Alternate horizontal and vertical strips, weaving them in an over-and-under pattern. Use shortest strips for edges of lattice.

Bake in the preheated oven for 15 minutes. Reduce oven temperature to 350 degrees F (175 degrees C). Continue baking until crust is golden brown, about 45 minutes more. Cool pie completely on a wire rack, about 2 hours.