Make the dough: Bring a medium pot of salted water to a boil. Add the spinach and cook until soft and bright, 3–5 minutes. Strain and rinse with cold water, then squeeze out as much moisture as possible. Set aside.
In a small food processor or chopping by hand, process the cooked spinach into a purée (you should have 3 tablespoons or 1¾ ounces; discard any excess).
On a clean work surface, mound the flour. Create a deep well in the center and add the eggs and spinach. Using a fork, beat the eggs and spinach. Then, starting from the interior wall of the well, gradually incorporate the flour into the wet ingredients as if scrambling eggs, until a thick batter-like consistency forms. Use a bowl scraper to cut and fold the remaining flour into the center until a shaggy mass forms; press it to form a dough.
With clean dry hands, move the dough and any remaining scraps and flour to a clean work surface. Sift some flour lightly over the board, then knead the dough until semi-smooth, about 3 minutes. Scrape the work surface of dried flakes and continue to knead dough gently until it is smooth and supple and springs back when poked, and most or all of the flour is incorporated, about 15 minutes more. (If dough is dry and tight, wrap in plastic and let rest 5–10 minutes before continuing to knead.) Wrap the ball tightly in plastic and refrigerate at least 1 hour or ideally 1 day.
Make the filling: In a fine sieve, drain the ricotta for 15 minutes. Meanwhile, in a meat grinder fitted with a fine plate, grind the mortadella, then pass it through the grinder a second time (alternatively, you can blend the mortadella with all the remaining ingredients in a food processor). Add the drained ricotta, Parmigiano, egg, and nutmeg; season with salt and mix until combined.
Pack the filling into a piping bag fitted with a round tip, and refrigerate until ready to use, up to 3 days.
Prepare the pasta: On a very lightly floured surface using a mattarello or rolling pin, roll out the dough as thinly as possible into a large circle at least 22 inches in diameter (see pg. 99). Use a sharp knife or bicycle cutter to portion dough into 2½-inch squares. Using the piping bag, place a scant teaspoon of filling onto the center of each square. Picking up the squares one at a time, bring two corners together to form a triangle, then press firmly and repeatedly to seal and thin all sides and create as large a border as possible around the filling. Fold one point toward you to form a channel along the triangle’s side, then fold the opposite point similarly and bring the tips together; press firmly to thin, flatten, and create a sealed circle at the base of the balanzoni.
Bring a large pot of water to boil. Once just about boiling, make the sauce: Set a large skillet over medium-high heat and melt the butter. Cook until the foam subsides and the milk solids begin to brown, 2–3 minutes. Add the sage, lower the heat to low, and cook until the butter is deep golden brown and the sage is crispy and fragrant, 1½–2 minutes. Remove from the heat and season with salt to taste.
Meanwhile, salt the boiling water and add the balanzoni; stir immediately. Cook until the pasta floats and is al dente, 1½–2 minutes. Strain thoroughly, then immediately add to the skillet with the butter; toss gently.
Transfer the pasta and sauce to a platter and serve immediately, sprinkled with more grated cheese.