Bring a pot of water to a rolling boil.
Salt generously, add the potatoes, and cook until tender but not falling apart, about 10 minutes.
Thirty seconds before the potatoes are done cooking, add the green beans to the pot.
Drain the potatoes and beans and set aside.
In the meantime, make the dressing by whisking together the mustard, vinegar, garlic, 1 Tbsp olive oil, the sugar, and ¼ tsp salt in a bowl. (Alternatively, combine the ingredients in a mason jar and shake until blended)
Taste and adjust if needed.
In a large skillet, sauté the dill in a splash of olive oil over medium-high heat.
Add a couple of pinches of salt, stir in the leek, and sauté until golden and slightly crispy (~ 4 to 5 minutes)
In a large bowl, gently toss the potatoes and green beans, celery, cucumber, chives, and half of the leek with most of the dressing.
Taste, and add a sprinkling of salt, if needed.
Turn out onto a platter and finish with a final drizzle of dressing and the remaining leek.
Serve chilled or at room temperature.