Heat grill to medium (350° to 450°) with an area left unheated for indirect heat. Toss potatoes in 1 tbsp. oil; grill flat side down until marks appear, 8 minutes. Turn over, moving to indirect area. Cook until tender, 10 minutes. Let cool, then cut into chunks.
Whisk remaining 3 tbsp. oil, mustard, vinegar, oregano, capers, chile flakes, and anchovies in a large bowl. Gently mix in remaining ingredients.
Note: Nutritional analysis is per serving.