In a large saucepan of boiling salted water, cook the potatoes until tender, about 15 minutes. Drain, shaking off the excess water. Transfer the potatoes to a bowl and coarsely mash.
Meanwhile, in a blender, puree the cucumber with the vinegar until smooth. Strain the cucumber vinegar into a bowl, pressing on the solids; discard the solids.
In a small skillet, heat the olive oil until shimmering. Add the shallots and cook over moderately low heat until softened, about 4 minutes. Add the cucumber vinegar and sugar, season with salt and pepper and bring to a simmer. Pour the cucumber vinaigrette over the potatoes. Add the chives and celery leaves, season with salt and pepper and stir gently to combine. Serve warm or at room temperature.
The potato salad can be refrigerated overnight.
One Serving 207 cal, 10 gm total fat, 1.4 gm sat fat, 27 gm carb, 3 gm fiber.