Fill a saucepan with water and bring to a boil. Add the green beans and kosher salt and blanch until crisp-tender, 2 to 3 minutes. Drain the beans in a colander, shaking off excess water. Set aside.
In a large nonstick skillet set over medium-high heat, add the hazelnuts and toast until fragrant, 3 to 5 minutes. Remove to a plate. In the same skillet, add the butter and cook until brown, about 1 minute. Add the shallots and saute until softened, about 3 minutes. Add the blanched green beans and cook, tossing, for 1 minute. Add the sherry vinegar and toasted nuts. Gently stir the beans and sprinkle with pepper.
Transfer the green beans to a serving plate and finish with the flaky sea salt and a drizzle of hazelnut oil.