To make the Braciole: Toast the panko bread crumbs in a dry skillet with a little olive oil over low heat, until golden. Add to a large mixing bowl along with the anchovy, garlic, bocconcini, red peppers, parsley, a drizzle of olive oil and some salt and pepper, to taste. Stir together until well combined.
Set the flank steak on a piece of plastic wrap. Make a deep horizontal slice along the steak almost all the way through and fan open like a book. Lay another piece of plastic wrap on top. Using the smooth side of a meat mallet, gently flatten the steak until about 1/2-inch thick; take care not to tear. Discard the top sheet of plastic wrap; rub the surface with olive oil and season with salt and pepper, to taste. Spread the stuffing evenly over the meat, leaving a 1-inch border all around. Arrange the eggs lengthwise down the center of the meat and roll up like a jelly roll log, using the plastic wrap as support. Tie the roll with kitchen twine in 4 to 5 places to secure - this will help hold the shape and keep the filling from falling out.
Preheat the oven to 350 degrees F. Put a roasting pan across 2 burners and heat over medium heat. Add a 3-count of olive oil and add the thyme and garlic. Cook for about a minute until fragrant. Carefully add the braciole and sear until evenly browned all over, approximately 2 minutes each side.
Add the sliced onions and bay leaves, then stir in the beef broth to deglaze. Add the canned tomatoes over the top, then nestle in the whole vine tomatoes around the braciole. Bring to a simmer, then cover with foil and put in the oven to braise for 45 to 60 minutes. When done, remove the foil and remove the braciole to a carving board to rest. Carefully remove the whole vine tomatoes, with a slotted spoon, to a plate. Let the sauce cool for about 5 minutes. Discard the thyme stems and bay leaf, then add the sauce to a blender and puree. Pour the sauce back into the pan and set over medium heat to bring to a simmer. Season with salt and pepper, to taste, and add the balsamic vinegar. Remove the kitchen twine from the beef and cut into 1-inch thick "pin-wheel" slices. Arrange the slices on a platter and arrange the whole vine tomatoes around the beef. Pour the sauce over the top, garnish with chopped parsley and serve.