whisk vinegar and mustard. Gradually whisk in oil. Mix in chives, shallot and garlic. Season to taste with salt and pepper.
Place potatoes in dry kitchen towel and squeeze to remove as much moisture as possible. Transfer potatoes to large bowl. Add salt and pepper and toss to combine.
Heat 1 tablespoon oil in heavy skillet over medium heat. Mound 1/3 cup potatoes in skillet. Using spatula, flatten to 3-inch round. Repeat 3 times, forming 4 pancakes.
Top each cake with 1/4 of goat cheese. Sprinkle each with 1 teaspoon chives.
Cover each with another 1/3 cup potatoes; press to adhere, enclosing cheese completely and flattening slightly. Cook until bottoms are golden. Add 1/2 tablespoon oil to skillet. Turn pancakes over; cook until golden. Toss green with enough dressing to coat.
Divide salad among plates. Arrange 1 pancake atop each salad.