Pizza Chiena (Easter Pie)

Fill a large bowl with ice water; set aside. Place six of the eggs in a medium pot and add cold water to cover by 1 inch. Bring to a boil over medium-high. Immediately remove from heat; cover and let stand 12 minutes. Immediately transfer eggs to prepared ice water; let cool about 10 minutes. Peel eggs and cut lengthwise into quarters, then again into 1/4-inch pieces. Transfer egg pieces to a small bowl. Cover and refrigerate until ready to use.

While eggs cook, pierce each sausage casing with a paring knife so the fat will render while they cook. Place 10 cups cold water, ham, sausages, and paprika in a large pot. Bring mixture to a boil over medium. Reduce heat to medium-low to maintain a simmer; simmer, undisturbed, until a thermometer inserted into thickest portion of sausages registers 160°F, about 30 minutes. Remove from heat. Using tongs, remove ham and sausages from pot and set aside until cool enough to handle, about 10 minutes. Ladle 1 cup cooking liquid from pot (along with any fat on top) into a medium-size metal bowl; refrigerate until chilled and fat solidifies at top of bowl, about 1 hour. (Discard remaining cooking liquid in pot.) Once meat is cool enough to handle, trim and discard all large fat pieces and sinew from ham; remove and discard casings from sausages. Cut meat roughly into 1/4-inch cubes. (You should have about 4 cups ham and 3 cups sausage.) Cover and refrigerate until ready to use.

Once cooking liquid has chilled long enough for the fat to separate, remove from refrigerator. Using a large spoon, skim off as much fat from the top as possible and place in a measuring cup. Add as much vegetable shortening to skimmed fat as needed to measure 1/2 cup total. Set aside. Set remaining chilled cooking liquid aside.

Beat flour, baking powder, and 1 teaspoon of the salt in a stand mixer fitted with a paddle attachment on medium speed until just combined, about 20 seconds. Slowly beat in reserved 1/2 cup fat mixture, little by little (about 1 tablespoon at a time). Reduce mixer speed to low and add 1/2 cup of the reserved chilled cooking liquid; beat until well incorporated and dough looks slightly shaggy and isn't sticky, 1 to 2 minutes. Discard remaining chilled cooking liquid.

Transfer dough to a clean, dry work surface. Knead using palms of your hands until well incorporated, about 1 minute. Dough should be soft but not sticky and hold together in 1 ball. Wrap dough ball in plastic wrap; let rest at room temperature 1 hour . (Dough may be tightly wrapped in plastic wrap and stored in refrigerator up to 2 days. Let come to room temperature before using.)

Whisk together four of the uncooked eggs in a large bowl. Add chopped ham and sausage, cooked egg pieces, mozzarella, provolone, parsley, pepper, and remaining 1 teaspoon salt to eggs; stir to combine. Set aside.

Preheat oven to 375°F with rack in lowest position. Coat a 10-inch springform pan (2 inches deep) with cooking spray. Unwrap dough on a clean, dry work surface. Divide dough evenly into 3 (about 6 1/2 ounces each) pieces. Roll 2 of the dough pieces together (about 13 ounces) into a 16-inch round. Carefully ease dough round into prepared springform pan, trying not to stretch dough too much; gently press dough into pan corners. Some of the dough should hang over pan sides by 1/4- to 1/2-inch. Spread meat-egg mixture evenly over dough in pan, packing it in firmly but gently. Whisk remaining egg in a small bowl. Using a pastry brush, brush the egg wash on the exposed edges of the bottom piece of dough, the part that's hanging over the pan. Reserve remaining egg wash.

Roll out remaining dough piece (about 6 1/2 ounces) into a 12-inch round. Place the 12-inch dough round on top of meat-egg mixture in pan. Trim edges of dough with scissors or a sharp paring knife so there's a 1/2-inch overhang all around. Gently roll the edge over on itself, the bottom dough over the top, toward the middle of the pan until it's tightly sealed. It should look like a rope and sit inside the edges of the pan (not on top of the rim).

Brush remaining egg wash over pie top. Make four (1-inch) slits into top of pie. Place pie on a large rimmed baking sheet and bake in preheated oven on bottom rack until pie top is golden brown and a thermometer inserted into center of pie registers 150°F, about 1 hour, 30 minutes. Transfer pie to a wire rack; let cool 15 minutes. Remove pie from pan. Let cool on wire rack 2 hours. Serve pie warm, at room temperature, or chilled, cut into wedges. Leftover pie may be tightly wrapped in plastic wrap and stored in refrigerator up to 5 days. Pie may also be wrapped in plastic wrap, placed in a sealed ziplock plastic freezer bag, and stored in freezer up to 3 months; let thaw in refrigerator, then reheat in a 350°F oven until hot and crispy.