Preheat the oven to 400 degrees F.
In a medium pot, combine the mirlitons, water, allspice, cinnamon stick, and salt, and bring to a boil. Reduce the heat and simmer until tender, about 8 minutes. Remove with a slotted spoon and drain in a colander. To the cooking liquid, add 1 cup of the sugar and lime juice and simmer until it thickens to a thick syrup. Remove from the heat and let cool.
Divide the piecrust dough in 1/2. Place 1 piece of dough on a lightly floured surface and roll out to an 11-inch circle, about 1/8-inch thick. Transfer to a 9-inch pie pan. Trim the crust with scissors or a sharp knife to within 1/2-inch of the outer rim. Return the cooked mirlitons to the pot, and stir to combine with the syrup. Pour into the piecrust. Roll out the remaining crust on a lightly floured surface. Place on top of the mirliton filling and tuck the overlapping crusts into the pan, forming a thick edge. Crimp the edges to seal and cut 1/2-inch long vents along the top crust.
With a pastry brush, brush the top of the crust with the cream, and sprinkle with the remaining 1/2 teaspoon of sugar. Bake for 10 minutes, then reduce the temperature to 375 degrees F. Bake until the crust is golden brown and the filling is bubbly, 35 to 40 minutes. Remove from the oven and cool on a wire rack for at least 1 hour before serving.
Serve warm or at room temperature.
Yield: 8 to 10 servings