Preheat oven to 450 degrees F (230 degrees C). Grease a large baking pan with olive oil.
Pierce potatoes and sweet potatoes all over with a fork. Place in the baking pan.
Bake in the preheated oven until tender, about 1 hour. Cut hot potatoes in half, holding them with a towel to protect your hands.
Dust a flat work surface with flour using a sifter. Press potatoes through a potato ricer directly onto the work surface. Sprinkle 1/4 cup Parmigiano-Reggiano cheese, salt, and pepper evenly over potatoes. Mix well.
Drizzle egg yolk over potato mixture; knead in by hand or with a bench scraper. Sift in flour gradually until potato mixture is smooth and slightly sticky, adding more flour as needed. Form into a ball; sprinkle flour over the top. Divide dough into 5 small balls. Dust each ball lightly with flour.
Roll each ball of dough into a 1/2-inch rope. Cut each rope into 1/2-inch gnocchi. Transfer gnocchi to a baking sheet lined with parchment paper. Repeat with remaining dough. Roll gnocchi over the tines of a fork or a gnocchi board to create ridges.
Bring a large pot of salted water to a rolling boil. Cook gnocchi in batches until they rise to the surface, 3 to 4 minutes per batch. Transfer to a baking pan using a slotted spoon. Let cool.
Cook 'nduja in a skillet over medium-low heat until it has a loose, spoonable consistency, 3 to 5 minutes. Add butter; cook until foamy. Add sage; cook until butter starts to brown and leaves are crisp, about 3 minutes. Remove from heat. Transfer sage to a plate and stir in balsamic vinegar. Let sauce cool slightly.
Spoon gnocchi into the skillet with the sauce. Cook over low heat, stirring constantly, until gnocchi are heated through and coated with sauce, about 5 minutes. Plate the gnocchi and top each serving with fried sage leaves, 1 tablespoon balsamic glaze, and 1 tablespoon Parmigiano-Reggiano cheese.