Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the broccolini, garlic, 2 tablespoons water, 1/2 teaspoon salt, and pepper to taste. Cover and cook, stirring occasionally, until the broccolini is tender, 8 to 10 minutes. Remove from the heat; keep warm.
Whisk the eggs, 2 tablespoons water, and salt and pepper to taste in a medium bowl. Heat the remaining 1 tablespoon olive oil in a medium nonstick skillet over medium-low heat. Add the egg mixture and cook, stirring gently with a rubber spatula, until just cooked through, 3 to 4 minutes.
Divide the cheese, broccolini and scrambled eggs among the rolls. Top with pepperoncini and drizzle with pepperoncini brine. Serve with chips.
Photograph by Christopher Testani
Recipe courtesy Food Network Magazine