Place the peeled, diced potatoes in a large pot of lightly salted boiling water and cook for 15–20 minutes, or until fork-tender.
Drain the potatoes well and return them to a large bowl; mash until very smooth using a potato masher or ricer.
Stir in 1 tsp salt, cayenne pepper, chopped tarragon, and 2 tbsp olive oil until evenly combined.
Add the shredded Parmesan, flour, and the egg yolk to the mashed potatoes. Mix with your hands until a soft, smooth dough forms, taking care not to overwork it.
Lightly flour a work surface and divide the dough into 3 portions. Roll each portion into a log about 1 inch (2.5 cm) thick, then cut into 1-inch (2.5 cm) pieces.
Bring a large pot of lightly salted water to a boil. Cook the gnocchi in batches until they rise to the surface, about 1 minute; remove with a slotted spoon and transfer to a bowl of ice water to stop cooking.
Drain the gnocchi well. Heat 2 tbsp olive oil in a frying pan over medium-high heat, add the gnocchi, and sauté gently for 5–6 minutes until golden and crisp on the outside.
Season with salt and pepper to taste and grate additional Parmesan over the top before serving warm.
