Place 3 lb. small red potatoes, quartered, in a large pot and pour in cold water to cover; season very generously with kosher salt. Bring to a boil, then reduce heat and simmer until potatoes are tender, 12–16 minutes.
Meanwhile, place 6 Persian cucumbers, trimmed, sliced ¼" thick, and 1 large red onion, finely chopped, in a large heatproof bowl.
Bring ⅔ cup distilled white vinegar, 3 Tbsp. sugar, 2 Tbsp. Dijon mustard, 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and ½ cup water to a simmer in a small saucepan over medium heat. Remove from heat and vigorously whisk in ⅔ cup extra-virgin olive oil. Immediately pour over cucumber mixture. Let sit, stirring occasionally to ensure vegetables are evenly submerged, at least 10 minutes or until potatoes are ready.
Drain potatoes and add to cucumber mixture; toss well to coat (it’s okay if some potatoes fall apart). Add ½ cup coarsely chopped dill, season with lots of freshly ground pepper, and gently mix to combine. Cover potato salad and chill until very cold, at least 2 hours and up to 2 days.
Just before serving, stir potato salad with a large spoon. Taste and season with more salt if needed. Transfer to a serving bowl and top with more dill and season with more pepper.
