Preheat a grill to medium-heat. Scatter the potatoes in an 8-inch disposable aluminum baking tray and toss with the garlic, thyme, lemon slices, a drizzle of EVOO and some salt and pepper. Cover the tray with aluminum foil and place on the grill until the potatoes are tender, about 30 minutes if placed directly on the grill or 40 minutes if on the upper rack.
While the potatoes are cooking, whisk together the remaining EVOO, lemon juice, vinegar, mustard, brown sugar and parsley in a large bowl. Season with salt and pepper and toss the potatoes in while they’re still warm, giving them a good stir to evenly coat. Serve the salad warm or at room temperature, garnished with scallions.