In a small bowl, whisk together 4 tbsp. oil, the vinegar, 3/4 tsp. salt, and 1/4 tsp. pepper and set aside.
Put celery leaves, celery, spinach, and parsley in a salad bowl.
Heat remaining tbsp. oil in a 12-in. frying pan over medium-high heat until shimmering. Add mushrooms and cook undisturbed until browned on underside, 1 to 2 minutes. Season with 1/4 tsp. salt and cook, stirring occasionally, until browned and tender but not too soft, 2 to 3 minutes.
Transfer mushrooms to bowl with celery, add vinaigrette and hazelnuts, and toss to coat. Scatter cheese over salad, season to taste with more salt and pepper, and toss again.