Preheat the oven to 400°F (200°C). Bake the russet potatoes directly on the rack (or a sheet pan) until a skewer slides in easily, 60–75 minutes.
As soon as they are just cool enough to handle, peel and pass through a potato ricer onto a clean work surface, spreading the potato to steam off excess moisture for 5 minutes.
Sprinkle the fine sea salt evenly over the potatoes, drizzle the large egg on top, and sift the 1.25 cups all-purpose flour over everything. Using a bench scraper and your hands, gently bring the mixture together, then knead very lightly just until a soft, smooth dough forms, 1–2 minutes; avoid overworking.
Dust the surface with the 0.5 cup all-purpose flour. Divide the dough into 4 pieces; roll each into a 3/4-inch-thick rope. Cut into 1-inch pieces and, if you like, roll each piece over a fork or gnocchi board to make ridges. Arrange on a floured, parchment-lined tray.
Bring the 4 quarts water to a gentle boil in a wide pot, then add the kosher salt.
In a large skillet over medium heat, melt the unsalted butter until it foams. Add the fresh sage leaves and cook, swirling, until the butter turns pale golden and the leaves are crisp, 3–5 minutes; remove from the heat.
Cook the gnocchi in batches: add a handful at a time to the boiling water; when they float, cook 30–60 seconds longer, then lift straight into the skillet. Add a splash of the cooking water and toss over low heat until the sauce lightly coats the gnocchi; season with the black pepper.
Divide among warm bowls and finish with Parmigiano Reggiano.