Trout Toast with Soft Scrambled Eggs

large eggs

kosher salt, plus more

unsalted butter, divided

(1"-thick) slices sourdough or country-style bread

crème fraîche or sour cream

skin-on, boneless smoked trout fillet (about 5 oz.), skin removed, flesh broken into 1" pieces

lemon, halved

Freshly ground black pepper

scallions, thinly sliced on a diagonal

coarsely chopped dill

mature arugula, tough stems trimmed (about 4 cups)

extra-virgin olive oil