Step 1

Cook potatoes: Add potatoes to a large pot of salted water. Bring to a boil and cook until fork-tender, about 12 minutes. (Don't overcook until soft, because they're getting grilled/cooked again). Drain well. Cut potatoes in half, then transfer to a large bowl.

Step 2

Preheat: Preheat grill over medium-high heat. (You can use a stovetop grill pan or barbecue, see note 2). Lightly brush grill with oil (see note 1).

Step 3

Make dressing: Meanwhile, in a small bowl, whisk together olive oil, mayonnaise, 2 tablespoons of the lemon juice, 1 tablespoon of the fresh thyme, grainy mustard, salt and pepper.

Step 4

Season and grill: Drizzle half the dressing over potatoes. Toss gently to coat. Add potatoes to the grill, and cook until golden brown all over and crisp on the edges, 5 to 10 minutes. Add the potatoes back to the large bowl. Drizzle with the remaining dressing. Toss to coat. Taste and add the remaining 1 tablespoon lemon juice, if needed. Add remaining 1 tablespoon thyme, green onions, fresh dill, and capers. Stir gently to mix.