Place the peeled and diced potatoes in a large pot of lightly salted boiling water and cook until fork-tender, about 15–20 minutes; drain well.
Return the drained potatoes to a large bowl and mash until smooth using a potato masher or ricer.
Stir in 1 tsp salt, cayenne pepper, chopped tarragon, and 2 tbsp olive oil until evenly distributed.
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, taking care 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 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 briefly transfer to an ice bath to stop cooking, then drain.
Heat a frying pan over medium-high heat with the remaining olive oil. Add the drained gnocchi and sauté for 5–6 minutes, stirring gently, until golden and crisp. Season with salt and pepper to taste.
Grate additional parmesan over the gnocchi and serve warm.