Heat a LARGE (12- to 13-inch) skillet over medium heat. Add 2 tablespoons olive oil and the mushrooms to the skillet and sauté until soft—about 10 minutes. Add the garlic and salt and pepper to taste. Sauté another 2 minutes, then remove from heat.
Lay the flank steak on a work surface and place a piece of plastic wrap over the top. Using a meat mallet or rolling pin, POUND the steak into a 9x12-inch rectangle—1/4-inch thick.
Lay the cheese over the steak, leaving a 2- to 3-inch area uncovered along one long edge. Next, lay the mushrooms over the cheese, followed by the prosciutto and arugula leaves.
Starting with the long edge covered with ingredients, carefully roll the flank steak. Tie the steak roll in 4 places with kitchen twine. Mix 1 teaspoon salt, 1/2 teaspoon pepper, paprika and flour and rub over the roll.
Preheat the oven to 350 degrees F. In a large skillet heat 2 tablespoons of oil over medium-high heat. Carefully place the steak roll in skillet and sear 5-8 minutes turning to brown on all sides.
Transfer the skillet to the oven and bake until cooked through and cheese is melted—30 minutes. Tent with foil and let the meat rest for 10 minutes, then carefully slice in 1/2-inch pieces.