Whisk the ingredients for the marinade together in a bowl. Add the beef and rub it into the meat. Cover and refrigerate for 20 minutes. (or longer if desired).
Heat a nonstick pan on medium heat and add half of the olive oil. Saute the onion and peppers and cook about 4-5 minutes, until they soften up.
Add the beef along with the marinade and cook another 5 minutes to nicely brown up the pieces.
Add the broth and tomato paste and cook another minute, stirring, until the mixture thins out and becomes a sauce. You can add more broth if needed. Reduce the heat and let it simmer for 15 - 20 minutes, until the beef is cooked through and tender. Taste and adjust the seasoning with salt as needed.
Heat the remaining oil as needed in a nonstick pan on medium heat. Warm the pita on both sides until golden brown then cut into wedges.