In a food processor, combine the pieces of cold butter, flour, Parmesan cheese, paprika, and ice water. Pulse 5 to 8 times or until the mixture begins to clump together. If necessary, add more ice water, 1/2 to 1 teaspoon at a time. Transfer to a floured surface and knead just until the dough holds together. Flatten it into a disk and wrap it in plastic wrap. Refrigerate for 20 minutes.
Heat the oven to 375 F.
Roll the chilled dough out to a 12-inch circle. Fit the crust into a 9 1/2 to 10-inch pie plate. Crimp the edges and line the pastry with a sheet of foil or parchment. Fill with pie weights or dry beans so the dough will not bubble up in the oven. Bake for 12 to 15 minutes. It should appear set and just beginning to brown.
Remove the pie weights and foil or paper and prick the crust with a fork. Return the crust to the oven and bake for 5 to 7 minutes longer.
Reduce the oven temperature to 350 F.
Melt the butter in a large skillet over medium heat. Sauté the mushrooms until golden brown. Add the diced chicken and green onions. Sauté for 3 to 5 minutes, or until the onion is softened.
Transfer the chicken and mushroom mixture to the cooled pie shell.
In a bowl, whisk eggs, salt and cream together. Pour the egg mixture over the chicken mixture.
Sprinkle cheese over top.
Bake for 45 to 55 minutes at 350 F or until set and golden brown. Begin checking the crust after about 20 minutes and protect the edge with a pie shield if necessary to prevent overbrowning.
Transfer the baked quiche to a rack and let it cool for 10 minutes before slicing.