Heat olive oil in a large frying pan over medium-high heat. Add onion and cook until lightly browned, about 3 minutes; add garlic and anchovy paste and cook 2 minutes more. Add kale and sauté until beginning to wilt. Stir in 1 cup chicken stock and cook until most of the liquid has evaporated and the greens are completely wilted, about 10 minutes.
Remove pan from heat and stir in cheese, currants, pine nuts, salt, and pepper.
Lay flank steak on a large piece of plastic wrap and top with a second piece of plastic wrap. Using a meat mallet or the bottom of a heavy skillet, pound beef to an even thickness of 3/8 in. Discard top sheet of plastic wrap. Spread filling on beef in a 1/2-in.-thick layer, leaving a 1/2-in. border on all sides.
Beginning with a long edge and using plastic wrap as a guide, roll beef around filling and place it on the counter, seam side down. Tie beef with butcher's twine at 1-in. intervals, then tie one string around the roll lengthwise so the ends are tucked in and the filling is contained. Sprinkle beef generously with salt and pepper.
Heat vegetable oil in a large high-sided frying pan over medium-high heat. Add beef and cook, turning, until well browned on all sides, about 8 minutes. Pour in wine and remaining 2 cups chicken stock, cover, lower heat until liquid in pot is gently simmering, and cook 13 minutes.
Transfer beef from pot to a cutting board, cut off butcher's twine, and cover with foil. Bring braising liquid to a boil over high heat and boil until reduced to 1/2 cup, about 15 minutes. Slice beef into 1/2-in. slices and serve with gravy.
Note: Nutritional analysis is per serving.