Place curry paste in a large saucepan and heat over medium-high heat. Break the paste apart using a wooden spoon and add about 1/4 of the coconut milk to the paste; cook and whisk until heated through and smooth, 2 to 4 minutes.
Stir chicken into coconut milk mixture and simmer for about 3 minutes; add the remaining coconut milk, fish sauce, brown sugar, and soy sauce and stir well. Turn heat to medium-low and simmer until chicken is no longer pink in the center and sauce is slightly thickened, about 30 minutes. Remove saucepan from heat and cool curry sauce slightly.
Mix eggplant, long beans, onion, bok choy, bamboo shoots, water chestnuts, garlic cloves, Thai dragon chile peppers, and basil to curry sauce. Return saucepan to burner and bring to a boil; reduce heat and simmer until vegetables are tender, about 30 minutes more.
Preheat oven to 375 degrees F (190 degrees C). Line a 2-quart, 8 1/2-inch baking dish with 1 pie crust.
Carefully transfer curry to the crust-lined baking dish until dish is filled to 1/2 inch from top. Moisten the exposed pie crust with water and lay the second pie crust over the filling, folding excess dough in a pleat-like formation. Press the 2 crusts together to form a seal. Cut a few slits into the top of the crust for ventilation.
Bake in the preheated oven for 15 minutes. Wrap the sealed edges of crust in aluminum foil and continue baking until top crust is golden, 30 to 45 minutes more. Cool pot pie slightly before serving.