Cook the potatoes in a saucepan of lightly salted boiling water for 10 minutes. Drain well and leave to dry for a few minutes.
Place 1 tablespoon of the oil in a large bowl with the sea salt and a generous grinding of pepper, add the potatoes and turn gently to coat in the oil.
Heat a large griddle pan over a medium heat, and when hot, add the potatoes in a single layer. Cook for 3 minutes on each side until golden. Remove and keep warm while you cook any remaining potatoes.
Whisk the remaining oil, mustard and tarragon together in a small bowl. Place the potatoes in a shallow bowl and pour over the dressing. Serve immediately.