Step 1

In a large saucepan, cover the potatoes with water and bring to a boil. Add a generous pinch of salt and simmer over moderately low heat until just tender, about 15 minutes. Drain the potatoes and let cool completely, then cut them in half lengthwise.

Step 2

Meanwhile, in a large bowl, whisk the crème fraîche, vinegar, corn syrup, mustard, celery seeds and garlic. Gradually whisk in the 1/4 cup of oil. Stir in the crushed red pepper, oregano and 1 teaspoon of minced parsley and season the dressing with salt and pepper.

Step 3

Light a grill or preheat a grill pan. Generously brush the potatoes with oil and season with salt and pepper. Grill the potatoes, cut side down, over moderately high heat, turning once, until lightly charred and hot, 5 minutes. Transfer to a plate and let cool slightly.

Step 4

Toss the potatoes with the dressing and season them with salt and pepper. Fold in the cucumber, red onion and the 1/2 cup of chopped parsley and serve.