Slowly whisk miso into warm water until smooth. Set aside.

Heat a deep, 12-inch sauté pan or very wide pot over medium heat, then add oil. Add shallot and ginger; cook, stirring occasionally, until shallot begins to brown. Add mushrooms and 1 teaspoon water, reduce heat to medium-low, and cover pan to allow mushrooms to sweat, about 5 minutes. Once mushrooms have begun to soften, remove lid, increase heat to medium, and cook until mushrooms brown, about 10 minutes. Scrape mixture into a bowl and set aside.

Drop green beans into the same pan, then pour in miso mixture. Increase heat a bit and cover to allow beans to steam. Stir periodically, replacing lid each time. Cook until beans are tender, about 10 minutes.

Add mushroom mixture to beans. Cook, stirring, until vegetables are heated through and miso mixture reduces to a glaze. Stir in lemon zest and serve.