Heat the oven to 350°F. Pat the beef chuck and pork shoulder dry; season all over with 1.5 teaspoons kosher salt and 0.5 teaspoon black pepper (you will use the remaining kosher salt later for the filling and the pasta water). Melt 2 tablespoons of the unsalted butter with the extra-virgin olive oil in a heavy Dutch oven over medium-high heat. Brown the meats on all sides, 6–8 minutes total. Add the yellow onion, carrot, and celery; cook, stirring, until softened, about 5 minutes. Add the rosemary sprig and bay leaf, pour in the dry white wine, and simmer 2 minutes to reduce slightly. Add the beef broth, bring to a simmer, cover, and braise in the oven until the meats are very tender, 1.5–2 hours.
Transfer the meats to a bowl to cool. Discard the rosemary and bay. Set the pot over medium heat and reduce the braising liquid to about 0.5 cup, 5–10 minutes, scraping up any browned bits; reserve this concentrated sugo for the filling and sauce.
Blanch the baby spinach in boiling water until just wilted, 30–45 seconds. Drain, rinse under cold water, squeeze completely dry, and chop very finely.
Make the pasta dough: On a work surface, mound the 00 flour and make a well. Add the large eggs and egg yolks; whisk with a fork, gradually drawing in flour to form a shaggy dough. Knead until smooth and elastic, 8–10 minutes, dusting the surface lightly with a little 00 flour only as needed. Wrap and let rest at room temperature for 30 minutes.
Make the filling: Finely chop the cooled meats and braised vegetables by hand (or pulse in a food processor) to a fine, cohesive mince. Scrape into a bowl and mix in the chopped spinach, Parmigiano Reggiano, ground nutmeg, and 2–3 tablespoons of the reserved sugo, until the mixture is spreadable but not wet. Taste and season with a little of the reserved salt if needed. Keep chilled while you roll the pasta.
Roll and form: Work with one third of the dough at a time, keeping the rest covered. Roll through a pasta machine to very thin sheets (setting 7 or 8), about 3 inches tall. Spoon or pipe a narrow line of filling along the long edge, leaving a 0.5-inch border. Fold the sheet over the filling to enclose, press to seal, then pinch firmly between your fingers at 1-inch intervals to create the “plin.” Use a fluted wheel or knife to cut each piece free, forming small rectangles. Arrange on a tray generously dusted with semolina flour. Repeat with remaining dough and filling.
Bring a large pot of water to a rolling boil and add the reserved salt. Boil the agnolotti until they float and the pasta is just tender, 2–3 minutes. Lift them with a spider directly into a wide skillet.
Finish and serve: Add the reserved 4 tablespoons butter and about 0.5 cup of the reserved sugo to the skillet. Swirl gently over low heat to coat the pasta and form a glossy sauce, 1–2 minutes, loosening with a splash of pasta water if needed. Serve immediately.