In a 4- to 5-quart pan over high heat, bring 2 quarts water to a boil. Add green beans and cook until bright green and tender-crisp, 1 to 2 minutes. Drain well and rinse with cold water until cool (or plunge in ice water); drain again.
In same pan, bring 2 quarts water to a boil over high heat. Add potato slices and cook until tender when pierced, 10 to 12 minutes; drain well and transfer to a large bowl.
Meanwhile, in a 1-cup glass measure, combine shallots and vinegar. In a 1- to 1 1/2-quart pan over medium heat, combine olive oil and summer savory. Stir occasionally until savory is limp and slightly darker and oil is fragrant, 10 to 12 minutes. If oil sizzles, reduce heat slightly.
Pour half the oil mixture into vinegar mixture and stir to combine; season with salt and pepper to taste. Pour over warm potatoes and mix gently to coat. Top potatoes with green beans and drizzle remaining oil mixture over beans. Serve warm or at room temperature.