Fill a medium skillet with enough water to reach a depth of half an inch. Bring to a boil, then add the green beans, lower the heat, cover and simmer, until crisp-tender, about 4 minutes. Drain in a colander and rinse with cold water until cool. Transfer to a serving dish.
Using the same skillet, melt the butter over medium heat. Add the scallion whites and mushrooms, season with salt and pepper and cook, stirring, until golden-brown, about 4 minutes. Stir in the flour for 1 minute (the mixture will be dry). Gradually stir in the chicken broth and cream until thickened. Season with salt and pepper, sprinkle with the scallion greens and remove from the heat.
Spoon the sauce over the green beans and sprinkle the potato chips on top.