Place peeled, diced potatoes in a large pot of lightly salted boiling water and cook 15–20 minutes, until fork-tender.
Drain the potatoes well and transfer to a large bowl; mash until smooth using a potato masher or ricer.
Stir in 1 tsp salt, cayenne pepper, chopped tarragon, and 2 tbsp olive oil until combined.
Add 1/2 cup shredded Parmesan, 1 1/3 cups plain flour, and the egg yolk; mix with your hands until a soft, smooth dough forms, being careful not to overwork it.
Lightly flour your work surface and divide the dough into three portions. Roll each into a log about 1 inch (2.5 cm) thick, then cut into 1-inch pieces.
Bring a large pot of lightly salted water to a boil. Cook the gnocchi in batches until they float to the surface, about 1 minute; remove with a slotted spoon and transfer to an ice bath to stop cooking, then drain.
Heat olive oil in a frying pan over medium-high heat. Drain the gnocchi and sauté in the hot pan for 5–6 minutes, stirring gently, until golden and crisp. Season with salt and pepper to taste.
Serve warm with extra grated Parmesan on top.