In a large bowl whisk together garlic, vinegar, Dijon, 2 tablespoons of the oil, oregano, salt, and pepper. Remove and reserve about 1/4 cup of the vinaigrette and set aside. Add steak to the remaining vinaigrette and turn to coat. Cover and refrigerate for at least 15 minutes and up to 1 hour.
Preheat the oven to the broiler setting.
Toss the broccolini in the remaining 2 tablespoons of olive oil and place on a baking sheet. Broil for 5 minutes and remove from the oven. Add beans to the broccolini and gently toss. Place a wire rack on top of the broccolini and bean mixture. Remove the skirt steak from the marinade and place on wire rack.
Broil the steak for 10 minutes (more or less depending on the desired doneness), turning halfway through. Remove steak from the rack and let rest for about 5 minutes. Meanwhile, transfer the broccolini mix to a medium mixing bowl. Toss with 1 tablespoon of the reserved vinaigrette.
Slice the steak and serve alongside broccolini. Drizzle with remaining vinaigrette and enjoy!