Preheat oven to 325 degrees F (165 degrees C). Line a baking sheet with aluminum foil; grease lightly.
Place pumpkin halves cut-side down on the prepared baking sheet.
Bake in the preheated oven until the fresh is tender when pierced with a fork, about 40 minutes. Cool until easily handled, about 15 minutes. Scoop pumpkin flesh out of the skin and into a food processor; puree until smooth.
Increase oven temperature to 450 degrees F (230 degrees C).
Whisk egg lightly in a large bowl. Stir in 1/2 cup brown sugar, 1 tablespoon flour, and salt. Add 1 3/4 cup pumpkin puree, evaporated milk, ginger, nutmeg, cloves, and cinnamon, mixing well after each addition.
Melt butter in a small saucepan over low heat. Stir in remaining 1 1/2 tablespoon flour until it forms a paste. Add remaining 1/4 cup brown sugar, white sugar, and water; bring to a boil. Reduce heat and simmer, stirring occasionally, until smooth, 3 to 5 minutes. Remove from heat.
Place bottom crust in a pie plate. Scatter some apples on top. Pour pumpkin puree mixture over apples. Arrange remaining apples in a circular pattern over puree. Moisten the rim of the pie with a small amount of water.
Cut top crust into 1/2-inch strips with a sharp knife or pastry wheel. Lay the 2 longest strips in an X in the center of the pie. Lay remaining strips on top, using the shortest ones on the edges, weaving them in an over-and-under pattern. Fold the ends under the edge of the bottom crust and flute the crust. Pour butter mixture gently over lattice.
Bake in the preheated oven for 12 minutes. Reduce temperature to 350 degrees F (175 degrees C) and continue baking until apples are soft and top is golden brown, 40 to 45 minutes. Transfer to a wire rack to cool before cutting.
