Trout Toast with Soft Scrambled Eggs

Name: Trout Toast with Soft Scrambled Eggs
Ingredients: 8 large eggs; 3/4 tsp. kosher salt, plus more; 6 Tbsp. unsalted butter, divided; 4 (1-thick) slices sourdough or country-style bread; 3 Tbsp. crème fraîche or sour cream; 1 skin-on, boneless smoked trout fillet (about 5 oz.), skin removed, flesh broken into 1 pieces; 1 lemon, halved; Freshly ground black pepper; 2 scallions, thinly sliced on a diagonal; 2 Tbsp. coarsely chopped dill; 4 oz. mature arugula, tough stems trimmed (about 4 cups); 2 tsp. extra-virgin olive oil