Place the peeled, diced potatoes in a large pot of lightly salted boiling water and cook 15–20 minutes until fork-tender.
Drain the potatoes and mash them in a large bowl until smooth using a potato masher or ricer.
Stir the 1 tsp salt, cayenne pepper, chopped tarragon, and 2 tbsp olive oil into the mashed potatoes until evenly combined.
Add the 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. Divide the dough into 3 portions and roll each into a log about 1 inch (2.5 cm) thick. Cut into 1-inch (2.5 cm) pieces.
Bring a pot of lightly salted water to a boil. Cook the gnocchi in batches; they are done when they rise to the surface, about 1 minute. Remove with a slotted spoon and transfer briefly to an ice-water bowl to stop cooking, then drain.
Heat 2 tbsp olive oil in a frying pan over medium-high heat. Add the drained gnocchi and sauté 5–6 minutes, stirring gently, until golden and crisp. Season with salt and pepper to taste.
Serve warm with extra grated parmesan sprinkled on top.