Cut fingerling potatoes in half, lengthwise. Coat with 1 tablespoon olive oil, 1/2 tsp salt and 1/2 tsp pepper. Place potatoes cut side up on a jelly roll pan.
Trim and snap green beans. Coat with 1/2 tablespoon olive oil, 1/4 tsp salt, and 1/4 tsp pepper. Place green beans on another jelly roll pan.
Bake potatoes in preheated oven for about 30 minutes, until soft and browned.
Halfway through the potato baking time, put in the green beans and roast those for about 15 minutes, until softened slightly.
While the potatoes and green beans are baking, make the tarragon cream sauce by combining all ingredients in a small bowl and whisking together.
Combine green beans and potatoes on a serving platter. Serve drizzled with tarragon cream dressing or serve it on the side.