To make the buttermilk-herb dressing, whisk together 6 Tbsp. buttermilk, 1/4 cup olive oil mayonnaise, 2 Tbsp. fresh lemon juice, 1 1/2 Tbsp. snipped fresh chives, 1 Tbsp. fresh flat-leaf parsley, 1/2 tsp. honey, and 1/4 tsp. each kosher salt and freshly ground black pepper in a bowl. Cover and refrigerate until ready to use. Makes 3/4 cup.
Combine eggs, dressing, mayonnaise, salt, and pepper in a bowl; mash with a fork. Fold in celery, fennel, and chives. Divide arugula between bread slices and top each with egg salad.