Bring potatoes, water to cover, and 1/4 cup of the salt to a boil in a large Dutch oven over medium-high. Reduce heat to medium-low, and cook until just fork-tender but not falling apart, about 8 minutes. Add haricots verts, and cook until beans are tender-crisp, about 1 minute. Drain well; cool completely, about 30 minutes. Transfer to a large bowl and cover with plastic wrap; chill until ready to use.
Whisk together lemon zest and lemon juice, shallot, vinegar, mustard, pepper, remaining 1 teaspoon salt, and 1 tablespoon of the chopped dill in a medium bowl; gradually add olive oil in a slow, steady stream, whisking until smooth.
Gently toss together potato-green bean mixture and 1/2 cup of the dressing, and let stand 30 minutes. Before serving, gently stir in radishes, remaining fresh dill, and remaining dressing. Sprinkle with chives.