Amazon is starting to roll out its redesigned Fire TV mobile app to customers, the company announced on Thursday. While you could previously only use the app as a backup remote, the updated experience lets you browse and discover content directly from your phone, manage your watchlist on the go, and play titles on your TV from the mobile app. “The updated app turns smartphones into a second screen for discovering what to watch next, making it easy to add a friend’s show recommendation to a watchlist even when away from home,” Amazon wrote in an email to TechCrunch. “This seamless integration between mobile and TV creates a more flexible and convenient viewing experience for Fire TV customers to find what they want to watch, fast.” The app is rolling out now in the U.S., Brazil, Canada, France, Germany, India, Italy, Japan, Mexico, Spain, and the U.K.…