In a large pot, add the potatoes, enough water to cover them by 2 inches and generously salt the water (like pasta!). Bring to a boil, then reduce the heat to medium-low and simmer for 10-15 minutes, or until easily pierced with a knife (this will vary greatly depending on the size of your potato). Drain the potatoes, add them to a large bowl, and set aside to cool off a bit.
While the potatoes cook, add the oil, vinegar, mustard, shallot, and ½ teaspoon salt to a jar with a tight-fitting lid. Seal and shake vigorously to combine. Alternatively, you can whisk everything together in a bowl.
Once the potatoes have cooled enough to handle but are still warm, slice them into thin rounds or half moons and add to a large bowl. Pour half of the dressing over the warm potatoes along with remaining ½ teaspoon salt and gently toss to coat. Let them hang out and soak up the dressing for about 5 minutes
Add in the cucumber, dill and remaining dressing. Toss everything together to coat. Taste and add more salt or vinegar if it tastes like it’s missing something. Top with more dill if you have some leftover and freshly grated black pepper.