Instead of topping this gnocchi with grated Parmesan, make little Parm crisps in the oven and use them as a garnish.
For the frico, preheat the oven to 325°. Line a baking sheet with parchment paper. Mound the cheese into 8 piles, spacing them evenly apart. Spread each pile into a thin, flat 3-inch circle. Bake until the frico are golden brown, 8 to 10 minutes. Let cool on the baking sheet until crisp, about 10 minutes.
For the pesto, in a large skillet, heat the olive oil over high until almost smoking. Add the Brussels sprouts, cut-side down. Cook until browned, about 3 minutes. Transfer to a food processor. Let cool slightly, about 5 minutes.
Add the spinach, cheese, pine nuts, and garlic to the processor bowl. Pulse until finely chopped, about 5 pulses. With the machine running, gradually add the EVOO, stopping to scrape down the sides of the bowl as needed. Season the pesto with salt.
For the gnocchi, in a large saucepan, cover the potatoes with cold water. Bring to a boil. Reduce heat to medium. Cook the potatoes until tender when pierced with a knife, about 25 minutes. Drain the potatoes and, using a potato ricer or masher, finely mash. Spread out the potatoes on a baking sheet. Let stand until cool enough to handle, about 10 minutes.
On a work surface, spread the potatoes into a large, flat mound. Drizzle the potatoes with the beaten egg. Sprinkle with 1 1/4 cups flour and the salt. Using a bench scraper, scrape and fold the potatoes until a slightly crumbly dough forms; sprinkle the potatoes with more flour if the dough gets too sticky. Knead the dough into a ball, then, using the bench scraper, cut into 8 pieces. On a lightly floured surface, roll each piece into a 3/4-inch-wide, 15-inch-long rope. Using the bench scraper, cut each rope into 3/4-inch pieces; dust the work surface with flour as needed to prevent sticking. Roll each piece gently over the tines of a fork to create ridges. Transfer the gnocchi to a lightly floured baking sheet.
Bring a large pot of water to a boil. Salt the water and, working in batches, cook the gnocchi until they float to the surface and are cooked through, 2 to 3 minutes per batch. Reserve 1/2 cup of the cooking water. Drain the gnocchi.
In a large bowl, toss the gnocchi with the pesto; add a little of the cooking water if they’re too dry. Divide the gnocchi among 4 bowls. Top each bowl with 2 frico.