Peel the potatoes and drop them in a bowl of cold water to keep them from turning brown.
Using the small julienne blade of a mandoline, cut the potatoes lengthwise in long matchsticks.
Place the potatoes on a clean kitchen towel and squeeze the towel to dry the potatoes without breaking them up.
Place the potatoes in a bowl and toss them with 1 teaspoon salt and ½ teaspoon pepper.
Heat 2 tablespoons oil in a small (8-inch) sauté pan with sloped sides over medium heat.
Add all the potatoes to the pan and arrange them so they lie flat.
Press them lightly with a metal spatula.
Allow the potatoes to cook undisturbed for 5 minutes.
Drizzle 1 tablespoon oil down the sides of the pan around the galette and cook for another 5 minutes.
When the galette is nicely browned on the bottom, loosen it carefully around the edge with a large, narrow, flexible metal spatula and flip it over with a large flat spatula.
Cook the galette for 2 minutes.
With the tip of a knife, run the pat of butter around the edges of the galette, allowing it to melt down the sides.
Continue to cook for 4 to 5 minutes, until nicely browned on the second side.
With the large flat spatula, transfer the galette to a paper towel and allow it to sit for 30 seconds.
Transfer to a board and cut in quarters with a large chef’s knife.
Transfer to a plate, sprinkle with chives and fleur de sel, and serve hot.
