Cook green beans in a pot of boiling salted water until bright green – about 2 mins. Drain and rinse under cold water to stop them cooking further. Set aside.
Heat oil in a frying pan on medium-high heat. Cook shallots/onion until frizzled and crisp. Transfer to paper towel to drain. Set aside.
Add mushrooms to the same pan and toss to coat with oil (add a touch more oil if you need to). Cook for a few minutes, until softened, stirring occasionally.
Reduce heat to medium and add green beans, butter, dill, ½ tsp sea salt. Cook, tossing often, spooning the foaming butter over the veg, until butter darkens slightly – about 4-5 mins.
Add vinegar and season with black pepper. Cook, stirring, for 1 minute.
Transfer the veg to a serving dish and spoon over any sauce from the pan. Top with crispy shallots/onions and grated parmesan cheese.
