Preheat the oven to 350˚. Place the rolls cut-side up on a baking sheet and bake until lightly toasted, about 5 minutes. Brush the insides with 2 tablespoons olive oil and season lightly with salt and pepper.
Whisk the eggs and 1/4 teaspoon salt in a medium bowl. Heat the remaining 1 tablespoon olive oil in a medium nonstick skillet over medium-low heat. Add the eggs and cook, stirring, until they just begin to scramble, about 30 seconds. Add the scallions and continue cooking, stirring, until just set, 4 to 5 minutes.
Spoon the eggs onto the roll bottoms and top with the caponata. Spread the ricotta on the roll tops and season with pepper. Toss the arugula, pepperoncini and brine in a bowl; season with salt and pepper and add to the sandwiches.