To make the dough: Sift together and then mound 3 cups of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs. Using a fork, beat together the eggs and begin to incorporate the flour, starting with the inner rim of the well. As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. The dough will come together when half of the flour is incorporated. Start kneading the dough with both hands, using the palms of your hands. Knead for about 15 minutes, adding any of the remaining four if necessary to create a cohesive mass. Once you have a cohesive mass, remove the dough from the board and scrape up and discard any leftover bits. Lightly reflour the board and continue kneading for 6 more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature.
For the filling: bring 6 quarts water to a rolling boil and add 2 tablespoons salt. In a 12-inch saucepan, add 1 tablespoon butter over high heat until hot but not smoking. Add the garlic and rosemary, and let cook until the garlic is light golden brown, about 5 minutes. Add the veal and pork meat, and brown on all sides, stirring occasionally, about 8 to 10 minutes. Season with salt and pepper, to taste. Do not be afraid to let the meat begin to caramelize a bit. Cook the spinach in boiling water for 1 minute. Drain well and add to the meat. Stir in the Parmigiano, 3 whole eggs, a pinch of nutmeg, and salt and pepper, to taste. Use a wooden spoon to mix until well combined. Set aside. Cut the pasta dough into 3 equally sized pieces. Re-wrap 2 of the pieces in plastic wrap and set aside. Begin working with the 1 unwrapped piece of dough. On a lightly floured work surface, use a floured rolling pin to roll out the pasta dough until it is 1/8-inch thick (you can also use a pasta machine and roll out the dough on its thinnest setting). Lay the resulting pasta sheet on a lightly floured surface with a long side facing you. Trim the edges so they are straight. Using a tablespoon, scoop equally sized spoonfuls of the filling and place along the bottom half of the pasta sheet, leaving a 1 1/2-inch border of dough at the bottom and sides: each dollop of filling should be approximately 1 1/2-inches away from the next. Pull the top edge of the pasta up and over the filling. The dough should form 1 large pocket over the dollops of filling. Seal the agnolotti by gently carefully molding the pasta over the filling and pressing lightly with your index finger to seal the edge of the dough to the pasta sheet; don't drag your finger along the dough to seal, or you risk ripping the dough. When it is sealed, there should be about 1/2-inch of excess dough visible along the bottom of the mounds of filling (where you sealed it). Be certain that you are sealing tightly while pressing out any pockets of air. Seal the left and right ends of the dough.
To shape agnolotti: Starting at 1 end of the dough, place the thumb and forefinger of each hand together as if you were going to pinch something and, leaving about 1-inch of space between your hands and holding your fingers vertically, pinch the filling in 1-inch increments, making about 3/4-inch of "pinched" are between each pocket of filling. It is important to leave this much "pinched" area between the agnolotti, or when the agnolotti are separated, they may come unsealed. Run a sharp knife or crimped pastry wheel along the bottom edge of the folded-over dough, separating the strip of filled pockets from the remainder of the pasta sheet. Don't cut too close to the filling, or you risk breaking the seal. Separate the individual agnolotti by cutting the center of each pinched area, rolling the pastry wheel away from you. Working quickly, place the agnolotti on a baking sheet dusted with a thin layer of cornmeal, which will help prevent sticking. Don't let the agnolotti touch each other, or they may stick together. Repeat with the 2 remaining dough balls until the entire bowl of filling has been used. Let the shaped agnolotti rest for 24 minutes. Bring 6 quarts water to a rolling boil, and add 2 tablespoons salt. Add the agnolotti to the water and cook until tender, about 4 minutes total. Drain well and toss with a sauce or ragu of your choice or a combination of beef broth and butter. Sprinkle with cheese and parsley and serve.