Place broccoli in a small saucepan with about an inch of salted water over medium-high heat. Bring to a boil, reduce heat, cover, and simmer until completely tender, about 15 minutes. Drain and immediately rinse with cold water. Drain completely.

Transfer broccoli and cheese to the bowl of a food processor and pulse until chopped, about 30 seconds. Add egg and process until smooth. If needed, add bread crumbs to thicken the mixture to a paste that holds it shape. Season with salt and pepper and pulse to combine.

Transfer broccoli mixture to a piping bag.

Divide pasta into four pieces, and, working with a piece at a time, roll into a strip about 3 inches wide and 16 inches long. Trim one long edge with a fluted cutter. Pipe filling onto the pasta sheet, leaving the ends bare.

Roll the pasta up and over the filling, making a tube filled with broccoli mixture. Press the ends closed. Use your fingers to press the pasta into small pillows, then use the fluted cutter to cut into individual pillows, and transfer to a lined baking sheet. Repeat with remaining pasta dough, folding any scraps into the next piece of dough.

Place the agnolotti in the freezer for 60 minutes, then transfer to a freezer bag for longer-term storage. Agnolotti can be cooked directly from the freezer.

Melt the butter in a large skillet over medium heat. Add onions and mushrooms, season with salt and pepper, and fry until mushrooms are browned in places. Reduce heat to keep warm.

Meanwhile, bring a large kettle of salted water to a simmer over medium-high heat. Add agnolotti and simmer until tender, about 5 minutes.

Use a slotted spoon to transfer to the skillet with the mushrooms. Increase heat, add about 1/4 cup of the pasta water, and simmer until everything is heated through.

Serve in warmed bowls, topped with additional grated cheese, if desired