Makes 8 servings.
The pickled cucumbers are also great on their own as a condiment for tuna-salad sandwiches.
6 tablespoons distilled white vinegar
4 teaspoons coarse kosher salt
2 1-pound English hothouse cucumbers, very thinly sliced
1/2 cup plus 3 tablespoons chopped fresh dill
3 1/4 pounds Yukon Gold potatoes (about 10 medium), unpeeled
Additional coarse kosher salt
1 cup very thinly sliced white onion
8 radishes, trimmed, thinly sliced
3/4 cup mayonnaise
Small radishes with green tops
Stir vinegar and 4 teaspoons coarse salt in small bowl until salt dissolves. Place cucumbers and 1/2 cup dill in heavy 1-gallon resealable plastic bag. Add vinegar mixture; seal bag. Turn several times to coat. Refrigerate overnight, turning bag occasionally. Pour cucumber mixture into large sieve set over bowl. Drain at least 1 hour and up to 3 hours. Discard brine.
Cook potatoes in large pot of boiling salted water until tender, about 30 minutes. Drain. Cool potatoes completely. Peel potatoes; quarter lengthwise. Cut crosswise into 1/2-inch-thick slices. Place potatoes in large bowl; sprinkle generously with coarse salt and pepper. Add drained cucumbers, onion, sliced radishes, and remaining 3 tablespoons dill; toss to blend. Let stand 1 hour. Stir mayonnaise into salad. Season generously with salt and pepper, if desired. (Salad can be made 1 day ahead. Cover and refrigerate.)
Mound salad in bowl; garnish with whole radishes. Serve cold or at room temperature.