To make dressing, whisk together balsamic vinegar, olive oil, oregano, salt and black pepper. Taste and adjust seasoning if necessary. Set aside.
In a large pot, bring salted water to a boil. Add the green beans and cook uncovered for 5-6 minutes or until just tender. Drain and immediately place them into ice-water for 2-3 minutes. Once cooled, drain the green beans and pat dry.
In a large bowl, toss the beans with half of the dressing and arrange on a serving platter. Layer the mushrooms on top. Sprinkle the sunflower kernels and parmesan cheese on top. Then drizzle with the remaining dressing.